Daily Papers

Novel view synthesis of dynamic scenes has been an intriguing yet challenging problem. Despite recent advancements, simultaneously achieving high-resolution photorealistic results, real-time rendering, and compact storage remains a formidable task. To address these challenges, we propose Spacetime Gaussian Feature Splatting as a novel dynamic scene representation, composed of three pivotal components. First, we formulate expressive Spacetime Gaussians by enhancing 3D Gaussians with temporal opacity and parametric motion/rotation. This enables Spacetime Gaussians to capture static, dynamic, as well as transient content within a scene. Second, we introduce splatted feature rendering, which replaces spherical harmonics with neural features. These features facilitate the modeling of view- and time-dependent appearance while maintaining small size. Third, we leverage the guidance of training error and coarse depth to sample new Gaussians in areas that are challenging to converge with existing pipelines. Experiments on several established real-world datasets demonstrate that our method achieves state-of-the-art rendering quality and speed, while retaining compact storage. At 8K resolution, our lite-version model can render at 60 FPS on an Nvidia RTX 4090 GPU.

Realistic animatable human avatars from monocular videos are crucial for advancing human-robot interaction and enhancing immersive virtual experiences. While recent research on 3DGS-based human avatars has made progress, it still struggles with accurately representing detailed features of non-rigid objects (e.g., clothing deformations) and dynamic regions (e.g., rapidly moving limbs). To address these challenges, we present STG-Avatar, a 3DGS-based framework for high-fidelity animatable human avatar reconstruction. Specifically, our framework introduces a rigid-nonrigid coupled deformation framework that synergistically integrates Spacetime Gaussians (STG) with linear blend skinning (LBS). In this hybrid design, LBS enables real-time skeletal control by driving global pose transformations, while STG complements it through spacetime adaptive optimization of 3D Gaussians. Furthermore, we employ optical flow to identify high-dynamic regions and guide the adaptive densification of 3D Gaussians in these regions. Experimental results demonstrate that our method consistently outperforms state-of-the-art baselines in both reconstruction quality and operational efficiency, achieving superior quantitative metrics while retaining real-time rendering capabilities. Our code is available at https://github.com/jiangguangan/STG-Avatar

The emergence of neural representations has revolutionized our means for digitally viewing a wide range of 3D scenes, enabling the synthesis of photorealistic images rendered from novel views. Recently, several techniques have been proposed for connecting these low-level representations with the high-level semantics understanding embodied within the scene. These methods elevate the rich semantic understanding from 2D imagery to 3D representations, distilling high-dimensional spatial features onto 3D space. In our work, we are interested in connecting language with a dynamic modeling of the world. We show how to lift spatio-temporal features to a 4D representation based on 3D Gaussian Splatting. This enables an interactive interface where the user can spatiotemporally localize events in the video from text prompts. We demonstrate our system on public 3D video datasets of people and animals performing various actions.

Reconstructing dynamic 3D scenes from 2D images and generating diverse views over time is challenging due to scene complexity and temporal dynamics. Despite advancements in neural implicit models, limitations persist: (i) Inadequate Scene Structure: Existing methods struggle to reveal the spatial and temporal structure of dynamic scenes from directly learning the complex 6D plenoptic function. (ii) Scaling Deformation Modeling: Explicitly modeling scene element deformation becomes impractical for complex dynamics. To address these issues, we consider the spacetime as an entirety and propose to approximate the underlying spatio-temporal 4D volume of a dynamic scene by optimizing a collection of 4D primitives, with explicit geometry and appearance modeling. Learning to optimize the 4D primitives enables us to synthesize novel views at any desired time with our tailored rendering routine. Our model is conceptually simple, consisting of a 4D Gaussian parameterized by anisotropic ellipses that can rotate arbitrarily in space and time, as well as view-dependent and time-evolved appearance represented by the coefficient of 4D spherindrical harmonics. This approach offers simplicity, flexibility for variable-length video and end-to-end training, and efficient real-time rendering, making it suitable for capturing complex dynamic scene motions. Experiments across various benchmarks, including monocular and multi-view scenarios, demonstrate our 4DGS model's superior visual quality and efficiency.

Asymptotic Safety provides a promising framework for the quantization of Scalar-Tensor theories as non-perturbatively renormalizable quantum field theories. Over the past years, growing evidence has accumulated for the existence of non-trivial renormalization group fixed points in a variety of Scalar-Tensor models. In this work, we investigate the robustness of these fixed points under significant modifications of the computational setup. In particular, we employ a gauge condition that explicitly depends on the dynamical scalar field, going beyond a mere gauge-parameter deformation of the standard linear covariant gauges commonly used in the literature. Exploring different field parameterizations, we identify fixed points belonging to the Gaussian Matter class in four spacetime dimensions that are consistent with previous studies. The real part of the critical exponents associated with the scalar-field mass and the non-minimal scalar-curvature coupling turn out to have very small absolute value. We provide a comprehensive exploration of how the anomalous dimension of the scalar field and field parameterization impact those critical exponents showing that fixed-point candidates with two or four relevant directions emerge.

Using a fully Bayesian approach, Gaussian Process regression is extended to include marginalisation over the kernel choice and kernel hyperparameters. In addition, Bayesian model comparison via the evidence enables direct kernel comparison. The calculation of the joint posterior was implemented with a transdimensional sampler which simultaneously samples over the discrete kernel choice and their hyperparameters by embedding these in a higher-dimensional space, from which samples are taken using nested sampling. Kernel recovery and mean function inference were explored on synthetic data from exoplanet transit light curve simulations. Subsequently, the method was extended to marginalisation over mean functions and noise models and applied to the inference of the present-day Hubble parameter, H_0, from real measurements of the Hubble parameter as a function of redshift, derived from the cosmologically model-independent cosmic chronometer and ΛCDM-dependent baryon acoustic oscillation observations. The inferred H_0 values from the cosmic chronometers, baryon acoustic oscillations and combined datasets are H_0= 66 pm 6, km,s^{-1},Mpc^{-1}, H_0= 67 pm 10, km,s^{-1},Mpc^{-1} and H_0= 69 pm 6, km,s^{-1},Mpc^{-1}, respectively. The kernel posterior of the cosmic chronometers dataset prefers a non-stationary linear kernel. Finally, the datasets are shown to be not in tension with ln R=12.17pm 0.02.

Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.

This paper studies a machine learning regression problem as a multivariate approximation problem using the framework of the theory of random functions. An ab initio derivation of a regression method is proposed, starting from postulates of indifference. It is shown that if a probability measure on an infinite-dimensional function space possesses natural symmetries (invariance under translation, rotation, scaling, and Gaussianity), then the entire solution scheme, including the kernel form, the type of regularization, and the noise parameterization, follows analytically from these postulates. The resulting kernel coincides with a generalized polyharmonic spline; however, unlike existing approaches, it is not chosen empirically but arises as a consequence of the indifference principle. This result provides a theoretical foundation for a broad class of smoothing and interpolation methods, demonstrating their optimality in the absence of a priori information.

Minkowski functionals quantify the morphology of smooth random fields. They are widely used to probe statistical properties of cosmological fields. Analytic formulae for ensemble expectations of Minkowski functionals are well known for Gaussian and mildly non-Gaussian fields. In this paper we extend the formulae to composite fields which are sums of two fields and explicitly derive the expressions for the sum of uncorrelated mildly non-Gaussian and Gaussian fields. These formulae are applicable to observed data which is usually a sum of the true signal and one or more secondary fields that can be either noise, or some residual contaminating signal. Our formulae provide explicit quantification of the effect of the secondary field on the morphology and statistical nature of the true signal. As examples, we apply the formulae to determine how the presence of Gaussian noise can bias the morphological properties and statistical nature of Gaussian and non-Gaussian CMB temperature maps.

We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic probability flow ODE decoder, is fundamentally flawed. It provably forces geodesics to decode as straight segments in data space, effectively ignoring any intrinsic data geometry beyond the ambient Euclidean space. Complementing this view, diffusion also admits a stochastic decoder via the reverse SDE, which enables an information geometric treatment with the Fisher-Rao metric. However, a choice of x_T as the latent representation collapses this metric due to memorylessness. We address this by introducing a latent spacetime z=(x_t,t) that indexes the family of denoising distributions p(x_0 | x_t) across all noise scales, yielding a nontrivial geometric structure. We prove these distributions form an exponential family and derive simulation-free estimators for curve lengths, enabling efficient geodesic computation. The resulting structure induces a principled Diffusion Edit Distance, where geodesics trace minimal sequences of noise and denoise edits between data. We also demonstrate benefits for transition path sampling in molecular systems, including constrained variants such as low-variance transitions and region avoidance. Code is available at: https://github.com/rafalkarczewski/spacetime-geometry.

Motivated by searches for weak primordial non-Gaussianity in the cosmic microwave background, we investigate perturbative departures from Gaussian stochastic gravitational wave background statistics using pulsar timing arrays. We consider a quadratic ansatz that shows three-point correlations at first order in the nonlinear coupling and covariance corrections at second order. We first examine the ansatz for gravitational wave helicity amplitudes. However, the timing residual bispectrum that this produces is not manifestly invariant under local polarization basis rotations for tensor and vector modes. Then, we formulate a polarization-frame-invariant phenomenological model for detector-projected amplitudes and evaluate its one-pulsar timing residual third moment with a simplified response. Our numerical benchmarks have shown that this gives a very small Gaussian-rms-normalized skewness.

The mean Hawking flux does not determine how quantum stress fluctuations perturb a black-hole geometry. That information resides in field correlations, including vacuum contributions absent from the outgoing luminosity. Here we connect the complete exterior Unruh Wightman function to a finite-band stochastic reconstruction of Schwarzschild geometry. We retain the incoming vacuum and both outgoing frequency branches, form the connected stress covariance before angular projection, and propagate its monopole through a density-driven radial response. A shared Gaussian reconstruction then resolves how the value and derivative of the lapse control different geometric observables. For the specified covariance and noise amplitude, 2,200 realizations give relative widths of 0.0185 for the local horizon radius and 0.0483 for the surface-gravity temperature parameter, consistently across histograms and reconstruction checks. Covariance derivatives explain this unequal sensitivity. The construction separates the quantum-state input from the gravitational closure and spatial sampling assumptions, providing a reproducible finite-resolution description of how Hawking-field correlations affect horizon positions and surface-gravity statistics.

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student t mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE.

Generative models and vision encoders have largely advanced on separate tracks, optimized for different goals and grounded in different mathematical principles. Yet, they share a fundamental property: latent space Gaussianity. Generative models map Gaussian noise to images, while encoders map images to semantic embeddings whose coordinates empirically behave as Gaussian. We hypothesize that both are views of a shared latent source, the Universal Normal Embedding (UNE): an approximately Gaussian latent space from which encoder embeddings and DDIM-inverted noise arise as noisy linear projections. To test our hypothesis, we introduce NoiseZoo, a dataset of per-image latents comprising DDIM-inverted diffusion noise and matching encoder representations (CLIP, DINO). On CelebA, linear probes in both spaces yield strong, aligned attribute predictions, indicating that generative noise encodes meaningful semantics along linear directions. These directions further enable faithful, controllable edits (e.g., smile, gender, age) without architectural changes, where simple orthogonalization mitigates spurious entanglements. Taken together, our results provide empirical support for the UNE hypothesis and reveal a shared Gaussian-like latent geometry that concretely links encoding and generation. Code and data are available https://rbetser.github.io/UNE/

This paper exposes the ergodic ceiling and thermodynamic inefficiency of current deep learning, which converges to a statistical average of historic human knowledge. True semantic novelty requires a path-dependent, spatiotemporally bounded observer (a Data LifeCone) to inject non-ergodic insight, achieving KL divergence and avoiding manifold lock-in. AI Safety must recognise that a mature Artificial Superintelligence (ASI) would regard human-AI symbiosis as a thermodynamic necessity to avoid model collapse. We therefore propose hard physical containment via a digital physics sandbox powered by a Holographic E8 Projection Engine to verify models against real-world constraints. Spacetime is modeled as an information substrate of nested face-centered cubic (FCC) lattices of oscillating Planck-scale spheres maximizing local information and entropy density. Cut-and-project methods from the E8 root lattice produce a quasi-crystalline geometry where tetrahedral voids support SU chiral structure and elastic-shear eigenvalues generate candidate mass spectra. Rest mass is treated as discrete, integer microstate counts on local holographic boundaries (Bekenstein bound), replacing floating-point approximations with strict integer arithmetic to provide an information-theoretic definition of matter. Stable particles emerge as recurring lattice dislocations, and continuum recovery proceeds via variational renormalisation-group flows and Fourier Neural Operators that learn continuous spectral operators to recover the Schrödinger equation as an emergent statistical description. Crucially, these top-down topological constraints offer a mechanism for "NP-to-P" complexity collapse: by restricting an algorithm's proposal space to physically conserved causal trajectories, the sandbox prunes the combinatorial tree to deterministic, polynomial-time paths.

We develop a spacetime neural network method with second order optimization for solving quantum dynamics from the high dimensional Schr\"{o}dinger equation. In contrast to the standard iterative first order optimization and the time-dependent variational principle, our approach utilizes the implicit mid-point method and generates the solution for all spatial and temporal values simultaneously after optimization. We demonstrate the method in the Schr\"{o}dinger equation with a self-normalized autoregressive spacetime neural network construction. Future explorations for solving different high dimensional differential equations are discussed.

Gaussian Processes (GPs) are a powerful tool for probabilistic modeling, but their performance is often constrained in complex, largescale real-world domains due to the limited expressivity of classical kernels. Quantum computing offers the potential to overcome this limitation by embedding data into exponentially large Hilbert spaces, capturing complex correlations that remain inaccessible to classical computing approaches. In this paper, we propose a Distributed Quantum Gaussian Process (DQGP) method in a multiagent setting to enhance modeling capabilities and scalability. To address the challenging non-Euclidean optimization problem, we develop a Distributed consensus Riemannian Alternating Direction Method of Multipliers (DR-ADMM) algorithm that aggregates local agent models into a global model. We evaluate the efficacy of our method through numerical experiments conducted on a quantum simulator in classical hardware. We use real-world, non-stationary elevation datasets of NASA's Shuttle Radar Topography Mission and synthetic datasets generated by Quantum Gaussian Processes. Beyond modeling advantages, our framework highlights potential computational speedups that quantum hardware may provide, particularly in Gaussian processes and distributed optimization.

3D Gaussian splatting (3DGS) has shown promising results in image rendering and surface reconstruction. However, its potential in volumetric reconstruction tasks, such as X-ray computed tomography, remains under-explored. This paper introduces R^2-Gaussian, the first 3DGS-based framework for sparse-view tomographic reconstruction. By carefully deriving X-ray rasterization functions, we discover a previously unknown integration bias in the standard 3DGS formulation, which hampers accurate volume retrieval. To address this issue, we propose a novel rectification technique via refactoring the projection from 3D to 2D Gaussians. Our new method presents three key innovations: (1) introducing tailored Gaussian kernels, (2) extending rasterization to X-ray imaging, and (3) developing a CUDA-based differentiable voxelizer. Experiments on synthetic and real-world datasets demonstrate that our method outperforms state-of-the-art approaches in accuracy and efficiency. Crucially, it delivers high-quality results in 4 minutes, which is 12times faster than NeRF-based methods and on par with traditional algorithms. Code and models are available on the project page https://github.com/Ruyi-Zha/r2_gaussian.

Given a stationary state-space model that relates a sequence of hidden states and corresponding measurements or observations, Bayesian filtering provides a principled statistical framework for inferring the posterior distribution of the current state given all measurements up to the present time. For example, the Apollo lunar module implemented a Kalman filter to infer its location from a sequence of earth-based radar measurements and land safely on the moon. To perform Bayesian filtering, we require a measurement model that describes the conditional distribution of each observation given state. The Kalman filter takes this measurement model to be linear, Gaussian. Here we show how a nonlinear, Gaussian approximation to the distribution of state given observation can be used in conjunction with Bayes' rule to build a nonlinear, non-Gaussian measurement model. The resulting approach, called the Discriminative Kalman Filter (DKF), retains fast closed-form updates for the posterior. We argue there are many cases where the distribution of state given measurement is better-approximated as Gaussian, especially when the dimensionality of measurements far exceeds that of states and the Bernstein-von Mises theorem applies. Online neural decoding for brain-computer interfaces provides a motivating example, where filtering incorporates increasingly detailed measurements of neural activity to provide users control over external devices. Within the BrainGate2 clinical trial, the DKF successfully enabled three volunteers with quadriplegia to control an on-screen cursor in real-time using mental imagery alone. Participant "T9" used the DKF to type out messages on a tablet PC.

Neural Radiance Fields (NeRFs) have demonstrated the remarkable potential of neural networks to capture the intricacies of 3D objects. By encoding the shape and color information within neural network weights, NeRFs excel at producing strikingly sharp novel views of 3D objects. Recently, numerous generalizations of NeRFs utilizing generative models have emerged, expanding its versatility. In contrast, Gaussian Splatting (GS) offers a similar render quality with faster training and inference as it does not need neural networks to work. It encodes information about the 3D objects in the set of Gaussian distributions that can be rendered in 3D similarly to classical meshes. Unfortunately, GS are difficult to condition since they usually require circa hundred thousand Gaussian components. To mitigate the caveats of both models, we propose a hybrid model Viewing Direction Gaussian Splatting (VDGS) that uses GS representation of the 3D object's shape and NeRF-based encoding of color and opacity. Our model uses Gaussian distributions with trainable positions (i.e. means of Gaussian), shape (i.e. covariance of Gaussian), color and opacity, and a neural network that takes Gaussian parameters and viewing direction to produce changes in the said color and opacity. As a result, our model better describes shadows, light reflections, and the transparency of 3D objects without adding additional texture and light components.

Do contemporary diffusion models preserve the class geometry of hyperspherical data? Standard diffusion models rely on isotropic Gaussian noise in the forward process, inherently favoring Euclidean spaces. However, many real-world problems involve non-Euclidean distributions, such as hyperspherical manifolds, where class-specific patterns are governed by angular geometry within hypercones. When modeled in Euclidean space, these angular subtleties are lost, leading to suboptimal generative performance. To address this limitation, we introduce HyperSphereDiff to align hyperspherical structures with directional noise, preserving class geometry and effectively capturing angular uncertainty. We demonstrate both theoretically and empirically that this approach aligns the generative process with the intrinsic geometry of hyperspherical data, resulting in more accurate and geometry-aware generative models. We evaluate our framework on four object datasets and two face datasets, showing that incorporating angular uncertainty better preserves the underlying hyperspherical manifold. Resources are available at: {https://github.com/IAB-IITJ/Harmonizing-Geometry-and-Uncertainty-Diffusion-with-Hyperspheres/}

In this note we explore the distribution of vacuum expectation values of the superpotential W_0 in explicit Type IIB flux compactifications. We show that the distribution can be approximated, universally across geometries, by a two-dimensional Gaussian with a model dependent standard deviation. We identify this behaviour in 20 Calabi-Yau orientifold compactifications with between two and five complex structure moduli by constructing a total of O(10^7) flux vacua. We observe a characteristic scaling behaviour of the width σ of our distributions with respect to the D3-charge contributions N_{flux} from fluxes which can be approximated by σsim N_{text{flux}}. This W_0 distribution implies that locating small values of |W_0| as a preferred regime associated with classes of string theory solutions typically featuring hierarchies, simplifies to the basic statement of finding small Euclidean norms of normally distributed values. We do also identify small modifications to this Gaussian behaviour in our samples which might be seen as indications for the breakdown of the continuous flux approximation commonly used in the context of statistical analyses of the flux landscape.

We derive the optically aligned system of image-scaling operators K_s that reproduces optical zoom (exact zoom-out, canonical zoom-in), by identifying a Huygens-like consistency principle in which each pixel acts as a secondary source. The system is determined in closed form by the camera's point spread function (PSF), itself the operator K_0. The operators compose, K_{s'} circ K_s = K_{ss'} (with band projection for down-up), and account for both upscaling and downscaling, which form a Moore-Penrose pseudoinverse pair, K_{1/s} = K_s^dagger. We show that image scaling, as routinely practiced, is neither optically faithful nor internally consistent; for example, common anti-aliasing rules generally correspond to no camera, and two resizes by a factor of 1/2 generally do not equal one resize by a factor of 1/4. Gaussian scale-space kernels cannot be exact finite-aperture PSFs. Gaussian scale-space is an idealized special case of what emerges here: a generalized scale-space for sampled images, derived entirely from camera optics, bidirectional, and not limited to Gaussian kernels. The system goes beyond scaling and supports composable and invertible subpixel geometric transformations. We empirically validate its optical predictions with real captures and noisy optics simulations, and demonstrate its practical impact on the stability of tasks such as super-resolution and classification.

The global topology of the Universe is a longstanding open question. In this work, we examine the statistical signatures of a positively curved universe with a Friedmann--Lemaitre--Robertson--Walker metric and the topology of a lens space L(p,q). Since these manifolds are generally statistically anisotropic and inhomogeneous, their cosmic microwave background (CMB) covariance matrices contain non-zero off-diagonal entries that depend on observer location. We compute these full harmonic-space covariance matrices for scalar perturbations for arbitrary lens spaces and observer position. Using the Kullback--Leibler divergence, we assess the distinguishability of these spaces from a simply connected three-sphere with the same curvature. The results show that topological signatures can still be significant in a cosmic-variance-limited regime even when the length, d_{rm NC}, of the shortest loop around the Universe through the observer exceeds the diameter, d_{rm LSS}, of the last-scattering surface by up to 10%. This distinguishability depends on the curvature, the lens space and the observer position mainly through the ratio d_{rm NC}/d_{rm LSS}, which is set not only by the curvature radius but also by p, q and the observer location. Therefore, for any value of the spatial curvature, however small, there are lens spaces and observer positions for which 0.985,d_{rm LSS}<d_{rm NC}<1.1,d_{rm LSS}. Such observers would find no matched circles in the CMB large enough to have been detected to date, yet the topology would remain potentially discoverable through its statistical signatures.

Physics simulation is paramount for modeling and utilizing 3D scenes in various real-world applications. However, integrating with state-of-the-art 3D scene rendering techniques such as Gaussian Splatting (GS) remains challenging. Existing models use additional meshing mechanisms, including triangle or tetrahedron meshing, marching cubes, or cage meshes. Alternatively, we can modify the physics-grounded Newtonian dynamics to align with 3D Gaussian components. Current models take the first-order approximation of a deformation map, which locally approximates the dynamics by linear transformations. In contrast, our GS for Physics-Based Simulations (GASP) pipeline uses parametrized flat Gaussian distributions. Consequently, the problem of modeling Gaussian components using the physics engine is reduced to working with 3D points. In our work, we present additional rules for manipulating Gaussians, demonstrating how to adapt the pipeline to incorporate meshes, control Gaussian sizes during simulations, and enhance simulation efficiency. This is achieved through the Gaussian grouping strategy, which implements hierarchical structuring and enables simulations to be performed exclusively on selected Gaussians. The resulting solution can be integrated into any physics engine that can be treated as a black box. As demonstrated in our studies, the proposed pipeline exhibits superior performance on a diverse range of benchmark datasets designed for 3D object rendering. The project webpage, which includes additional visualizations, can be found at https://waczjoan.github.io/GASP.

Recently, 3D Gaussian splatting (3D-GS) has achieved great success in reconstructing and rendering real-world scenes. To transfer the high rendering quality to generation tasks, a series of research works attempt to generate 3D-Gaussian assets from text. However, the generated assets have not achieved the same quality as those in reconstruction tasks. We observe that Gaussians tend to grow without control as the generation process may cause indeterminacy. Aiming at highly enhancing the generation quality, we propose a novel framework named GaussianDreamerPro. The main idea is to bind Gaussians to reasonable geometry, which evolves over the whole generation process. Along different stages of our framework, both the geometry and appearance can be enriched progressively. The final output asset is constructed with 3D Gaussians bound to mesh, which shows significantly enhanced details and quality compared with previous methods. Notably, the generated asset can also be seamlessly integrated into downstream manipulation pipelines, e.g. animation, composition, and simulation etc., greatly promoting its potential in wide applications. Demos are available at https://taoranyi.com/gaussiandreamerpro/.

Learning in Gaussian Process models occurs through the adaptation of hyperparameters of the mean and the covariance function. The classical approach entails maximizing the marginal likelihood yielding fixed point estimates (an approach called Type II maximum likelihood or ML-II). An alternative learning procedure is to infer the posterior over hyperparameters in a hierarchical specification of GPs we call Fully Bayesian Gaussian Process Regression (GPR). This work considers two approximation schemes for the intractable hyperparameter posterior: 1) Hamiltonian Monte Carlo (HMC) yielding a sampling-based approximation and 2) Variational Inference (VI) where the posterior over hyperparameters is approximated by a factorized Gaussian (mean-field) or a full-rank Gaussian accounting for correlations between hyperparameters. We analyze the predictive performance for fully Bayesian GPR on a range of benchmark data sets.

Partial differential equations (PDEs) are important tools to model physical systems, and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works like a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDE, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.

Weather nowcasting is an essential task that involves predicting future radar echo sequences based on current observations, offering significant benefits for disaster management, transportation, and urban planning. Current prediction methods are limited by training and storage efficiency, mainly focusing on 2D spatial predictions at specific altitudes. Meanwhile, 3D volumetric predictions at each timestamp remain largely unexplored. To address such a challenge, we introduce a comprehensive framework for 3D radar sequence prediction in weather nowcasting, using the newly proposed SpatioTemporal Coherent Gaussian Splatting (STC-GS) for dynamic radar representation and GauMamba for efficient and accurate forecasting. Specifically, rather than relying on a 4D Gaussian for dynamic scene reconstruction, STC-GS optimizes 3D scenes at each frame by employing a group of Gaussians while effectively capturing their movements across consecutive frames. It ensures consistent tracking of each Gaussian over time, making it particularly effective for prediction tasks. With the temporally correlated Gaussian groups established, we utilize them to train GauMamba, which integrates a memory mechanism into the Mamba framework. This allows the model to learn the temporal evolution of Gaussian groups while efficiently handling a large volume of Gaussian tokens. As a result, it achieves both efficiency and accuracy in forecasting a wide range of dynamic meteorological radar signals. The experimental results demonstrate that our STC-GS can efficiently represent 3D radar sequences with over 16times higher spatial resolution compared with the existing 3D representation methods, while GauMamba outperforms state-of-the-art methods in forecasting a broad spectrum of high-dynamic weather conditions.

Ensembles of neural network weight matrices are studied through the training process for the MNIST classification problem, testing the efficacy of matrix models for representing their distributions, under assumptions of Gaussianity and permutation-symmetry. The general 13-parameter permutation invariant Gaussian matrix models are found to be effective models for the correlated Gaussianity in the weight matrices, beyond the range of applicability of the simple Gaussian with independent identically distributed matrix variables, and notably well beyond the initialisation step. The representation theoretic model parameters, and the graph-theoretic characterisation of the permutation invariant matrix observables give an interpretable framework for the best-fit model and for small departures from Gaussianity. Additionally, the Wasserstein distance is calculated for this class of models and used to quantify the movement of the distributions over training. Throughout the work, the effects of varied initialisation regimes, regularisation, layer depth, and layer width are tested for this formalism, identifying limits where particular departures from Gaussianity are enhanced and how more general, yet still highly-interpretable, models can be developed.

A neural architecture with randomly initialized weights, in the infinite width limit, is equivalent to a Gaussian Random Field whose covariance function is the so-called Neural Network Gaussian Process kernel (NNGP). We prove that a reproducing kernel Hilbert space (RKHS) defined by the NNGP contains only functions that can be approximated by the architecture. To achieve a certain approximation error the required number of neurons in each layer is defined by the RKHS norm of the target function. Moreover, the approximation can be constructed from a supervised dataset by a random multi-layer representation of an input vector, together with training of the last layer's weights. For a 2-layer NN and a domain equal to an n-1-dimensional sphere in {mathbb R}^n, we compare the number of neurons required by Barron's theorem and by the multi-layer features construction. We show that if eigenvalues of the integral operator of the NNGP decay slower than k^{-n-2{3}} where k is an order of an eigenvalue, then our theorem guarantees a more succinct neural network approximation than Barron's theorem. We also make some computational experiments to verify our theoretical findings. Our experiments show that realistic neural networks easily learn target functions even when both theorems do not give any guarantees.

Faithful visualizations of data residing on manifolds must take the underlying geometry into account when producing a flat planar view of the data. In this paper, we extend the classic stochastic neighbor embedding (SNE) algorithm to data on general Riemannian manifolds. We replace standard Gaussian assumptions with Riemannian diffusion counterparts and propose an efficient approximation that only requires access to calculations of Riemannian distances and volumes. We demonstrate that the approach also allows for mapping data from one manifold to another, e.g. from a high-dimensional sphere to a low-dimensional one.

Variational inference (VI) seeks to approximate a target distribution pi by an element of a tractable family of distributions. Of key interest in statistics and machine learning is Gaussian VI, which approximates pi by minimizing the Kullback-Leibler (KL) divergence to pi over the space of Gaussians. In this work, we develop the (Stochastic) Forward-Backward Gaussian Variational Inference (FB-GVI) algorithm to solve Gaussian VI. Our approach exploits the composite structure of the KL divergence, which can be written as the sum of a smooth term (the potential) and a non-smooth term (the entropy) over the Bures-Wasserstein (BW) space of Gaussians endowed with the Wasserstein distance. For our proposed algorithm, we obtain state-of-the-art convergence guarantees when pi is log-smooth and log-concave, as well as the first convergence guarantees to first-order stationary solutions when pi is only log-smooth.

Disc winds are critical components of accreting astrophysical systems and their observations can yield a wide variety of spectral line profiles. Yet, inferring disc wind parameters remains challenging: the underlying parameter spaces are high-dimensional and degenerate, while the radiative-transfer simulations are imperfect and computationally expensive. This paper tackles bridging this gap. Here, we present 'Speculate', an open-source framework for rapidly emulating and fitting observations with synthetic accretion disc wind spectra, generated from the Monte Carlo radiative-transfer code Sirocco. Speculate includes the release of our cataclysmic variable (CV) training data, a library of 88,680 sparsely gridded Sirocco spectra spanning the ultraviolet and optical wave bands, a suite of pre-trained emulators, and its interactive Marimo user interface. Speculate compresses spectral grids using principal component analysis and then provides three approaches to interpolate the resulting latent representation: with a regular-grid interpolation, a neural network, or a Gaussian process. All three architectures enable Speculate to generate a Sirocco spectrum between 1-40,ms, thereby enabling efficient inference. The first two support rapid χ^2 optimisation. The Gaussian process utilises a Bayesian inference engine coupled with a covariance-aware, flexible likelihood function, adapted and accelerated from the Starfish framework. This function helps absorb correlated discrepancies between simulations and data. Tests show all architectures reproduce unseen Sirocco spectra with broadly comparable accuracy and can recover their physical parameters. Subsequently, we apply Speculate to 5 representative nova-like CV spectra, reproducing several observed line features while also exposing mismatches where the current model grid omits relevant physics.

A variety of infinitely wide neural architectures (e.g., dense NNs, CNNs, and transformers) induce Gaussian process (GP) priors over their outputs. These relationships provide both an accurate characterization of the prior predictive distribution and enable the use of GP machinery to improve the uncertainty quantification of deep neural networks. In this work, we extend this connection to neural operators (NOs), a class of models designed to learn mappings between function spaces. Specifically, we show conditions for when arbitrary-depth NOs with Gaussian-distributed convolution kernels converge to function-valued GPs. Based on this result, we show how to compute the covariance functions of these NO-GPs for two NO parametrizations, including the popular Fourier neural operator (FNO). With this, we compute the posteriors of these GPs in regression scenarios, including PDE solution operators. This work is an important step towards uncovering the inductive biases of current FNO architectures and opens a path to incorporate novel inductive biases for use in kernel-based operator learning methods.

In this manuscript we consider the problem of generalized linear estimation on Gaussian mixture data with labels given by a single-index model. Our first result is a sharp asymptotic expression for the test and training errors in the high-dimensional regime. Motivated by the recent stream of results on the Gaussian universality of the test and training errors in generalized linear estimation, we ask ourselves the question: "when is a single Gaussian enough to characterize the error?". Our formula allow us to give sharp answers to this question, both in the positive and negative directions. More precisely, we show that the sufficient conditions for Gaussian universality (or lack of thereof) crucially depend on the alignment between the target weights and the means and covariances of the mixture clusters, which we precisely quantify. In the particular case of least-squares interpolation, we prove a strong universality property of the training error, and show it follows a simple, closed-form expression. Finally, we apply our results to real datasets, clarifying some recent discussion in the literature about Gaussian universality of the errors in this context.

The Helmholtz equation governs time-harmonic wave propagation, and in dissipative media a complex modulus renders its squared wavenumber κ^2 complex. Inferring such fields from sparse, noisy data calls for solvers that also quantify their own uncertainty. Physics-informed Gaussian-process (GP) regression supplies this by returning a posterior over the solution, yet operator-conditioned formulations have been developed almost exclusively for real-valued fields. We extend operator-informed GP regression to complex-valued Helmholtz problems by realifying the complex operator into an equivalent coupled real block, which enables inference with standard real-valued GP conditioning. The construction admits a family of priors, from a proper diagonal prior to coregionalized and multiscale variants, and conditions on PDE residuals and boundary traces. On benchmark problems in one to three dimensions, the solver is competitive with finite-difference and neural-network baselines at a far smaller interior-constraint budget. Unlike those deterministic baselines, it returns a posterior over the complex wavefield rather than a point estimate. Applied to in vivo brain magnetic resonance elastography, a proper multiscale prior reconstructs the shear curl field to a correlation of 0.77 with measurement, above a 0.75 target. The gain arises from the multiscale kernel rather than from real--imaginary coupling. We further identify a low-frequency accuracy ceiling set by model mismatch and a posterior uncertainty that is not yet calibrated. Calibrated uncertainty therefore emerges as the central next step for probabilistic wavefield inference in dissipative media.

For robots to robustly understand and interact with the physical world, it is highly beneficial to have a comprehensive representation - modelling geometry, physics, and visual observations - that informs perception, planning, and control algorithms. We propose a novel dual Gaussian-Particle representation that models the physical world while (i) enabling predictive simulation of future states and (ii) allowing online correction from visual observations in a dynamic world. Our representation comprises particles that capture the geometrical aspect of objects in the world and can be used alongside a particle-based physics system to anticipate physically plausible future states. Attached to these particles are 3D Gaussians that render images from any viewpoint through a splatting process thus capturing the visual state. By comparing the predicted and observed images, our approach generates visual forces that correct the particle positions while respecting known physical constraints. By integrating predictive physical modelling with continuous visually-derived corrections, our unified representation reasons about the present and future while synchronizing with reality. Our system runs in realtime at 30Hz using only 3 cameras. We validate our approach on 2D and 3D tracking tasks as well as photometric reconstruction quality. Videos are found at https://embodied-gaussians.github.io/.

In the past two years, several points of view have been proposed to address the question of the generalization of the theory of free probability to random tensors with different invariances, and it is unclear at this point whether they lead to the same notions of tensorial free cumulants and freeness. One way to approach this problem, developed by Collins, Gurau and the second named author for local unitary invariant random tensors, relies on finite size quantities involving averages over the invariance group, and whose asymptotics naturally possess the properties expected for tensorial generalizations of free cumulants of arbitrary orders. At this point, this approach has only been carried out for certain distributions, and for a subset of the moments that define such theories, and a more systematic and exhaustive study is lacking. This is the program initiated in this paper: we link this approach to the one proposed by Nechita and Park; extend a number of their results as well as those of the aforementioned paper to arbitrary orders of fluctuations, thereby generalizing higher order free cumulants; push further the study of distributions with larger invariance groups; detail the link with the asymptotics of the free-energies of the tensor HCIZ and BGW integrals; and provide formulae for tensorial free cumulants of products of tensors. Another important question is that of the definition of concrete distributions whose tensorial free-cumulants take non-trivial values. We compute the tensorial free cumulants for Gaussian random tensors with non-trivial covariances, and show that they provide such examples.

This paper studies the scale and redshift variation of density and velocity distributions in self-gravitating collisionless dark matter flow by a halo-based non-projection approach. All particles are divided into halo and out-of-halo particles for redshift variation of distributions. Without projecting particle fields onto a structured grid, the scale variation is analyzed by identifying all particle pairs on different scales r. We demonstrate that: i) Delaunay tessellation can be used to reconstruct the density field. The density correlation, spectrum, and dispersion functions were obtained, modeled, and compared with the N-body simulation; ii) the velocity distributions are symmetric on both small and large scales and are non-symmetric with a negative skewness on intermediate scales due to the inverse energy cascade at a constant rate varepsilon_u; iii) On small scales, the even order moments of pairwise velocity Delta u_L follow a two-thirds law (-varepsilon_ur)^{2/3}, while the odd order moments follow a linear scaling langle(Delta u_L)^{2n+1}rangle=(2n+1)langle(Delta u_L)^{2n}ranglelangleDelta u_Lrangler; iv) The scale variation of the velocity distributions was studied for longitudinal velocities u_L or u_L^{'}, pairwise velocity (velocity difference) Delta u_L=u_L^{'}-u_L and velocity sum Sigma u_L=u^{'}_L+u_L. Fully developed velocity fields are never Gaussian on any scale, despite that they can initially be Gaussian; v) On small scales, u_L and Sigma u_L can be modeled by a X distribution to maximize the system entropy; vi) On large scales, Delta u_L and Sigma u_L can be modeled by a logistic or a X distribution; vii) the redshift variation of the velocity distributions follows the evolution of the X distribution involving a shape parameter alpha(z) decreasing with time.

A model involving Gaussian processes (GPs) is introduced to simultaneously handle multi-task learning, clustering, and prediction for multiple functional data. This procedure acts as a model-based clustering method for functional data as well as a learning step for subsequent predictions for new tasks. The model is instantiated as a mixture of multi-task GPs with common mean processes. A variational EM algorithm is derived for dealing with the optimisation of the hyper-parameters along with the hyper-posteriors' estimation of latent variables and processes. We establish explicit formulas for integrating the mean processes and the latent clustering variables within a predictive distribution, accounting for uncertainty on both aspects. This distribution is defined as a mixture of cluster-specific GP predictions, which enhances the performances when dealing with group-structured data. The model handles irregular grid of observations and offers different hypotheses on the covariance structure for sharing additional information across tasks. The performances on both clustering and prediction tasks are assessed through various simulated scenarios and real datasets. The overall algorithm, called MagmaClust, is publicly available as an R package.

We introduce HyperGaussians, a novel extension of 3D Gaussian Splatting for high-quality animatable face avatars. Creating such detailed face avatars from videos is a challenging problem and has numerous applications in augmented and virtual reality. While tremendous successes have been achieved for static faces, animatable avatars from monocular videos still fall in the uncanny valley. The de facto standard, 3D Gaussian Splatting (3DGS), represents a face through a collection of 3D Gaussian primitives. 3DGS excels at rendering static faces, but the state-of-the-art still struggles with nonlinear deformations, complex lighting effects, and fine details. While most related works focus on predicting better Gaussian parameters from expression codes, we rethink the 3D Gaussian representation itself and how to make it more expressive. Our insights lead to a novel extension of 3D Gaussians to high-dimensional multivariate Gaussians, dubbed 'HyperGaussians'. The higher dimensionality increases expressivity through conditioning on a learnable local embedding. However, splatting HyperGaussians is computationally expensive because it requires inverting a high-dimensional covariance matrix. We solve this by reparameterizing the covariance matrix, dubbed the 'inverse covariance trick'. This trick boosts the efficiency so that HyperGaussians can be seamlessly integrated into existing models. To demonstrate this, we plug in HyperGaussians into the state-of-the-art in fast monocular face avatars: FlashAvatar. Our evaluation on 19 subjects from 4 face datasets shows that HyperGaussians outperform 3DGS numerically and visually, particularly for high-frequency details like eyeglass frames, teeth, complex facial movements, and specular reflections.

Non-linear interactions during inflation generate non-Gaussianities in the distribution of primordial curvature. In many theories, the physics is scale-invariant, such that the induced three-point function depends solely on a dimensionless shape function S(x,y)sim k^6B_ζ(kx,ky,k). To confront such models with observations, one typically builds specialized estimators for each shape, then applies them to cosmic microwave background datasets at significant computational expense. In this Letter, we take a different approach, directly reconstructing S(x,y) from observations using an efficient logarithmically-binned estimator in primordial-space (motivated by the modal program). Applying this to temperature and polarization maps from Planck, we obtain high-resolution shape measurements across the full (x,y)-plane, including squeezed limits. Our approach is close-to-optimal, highly interpretable, and preserves the information content on (optimally-analyzed) standard templates within approx 10%; moreover, we can use it to assess the scale-dependence of our constraints, finding that Planck is sensitive to approx 6 e-folds of non-Gaussian evolution with a peak sensitivity around 0.1h,Mpc^{-1}. Since we work directly in shape-space, data and theory can be compared in milliseconds. As an example, we perform a search for massive particle exchange using a suite of over 20,000 theoretical templates computed with exact bootstrap methods (for the first time) across a wide range of masses, spins, and sound-speeds; the spin-two analysis yields a maximum significance of 2.6σ. Our approach can be used to probe a wide range of scale-invariant models in orders-of-magnitude less time than with direct estimators, allowing the inflationary paradigm to be explored in new ways.

Large nonlinear recurrent neural networks with random couplings generate high-dimensional, potentially chaotic activity whose structure is of interest in neuroscience and other fields. A fundamental object encoding the collective structure of this activity is the N times N covariance matrix. Prior analytical work on the covariance matrix has been limited to low-dimensional summary statistics. Recent work proposed an ansatz in which, at large N, the covariance matrix for a typical quenched realization takes the same form as that of a linear network with the same couplings, driven by independent noise, with DMFT order parameters setting the transfer function and the noise spectrum. Here, we derive this ansatz using the two-site cavity method, providing two derivations with complementary perspectives. The first decomposes each unit's activity into a linear response to its local field and a nonlinear residual, and shows that cross-covariances between residuals at distinct sites are strongly suppressed, so the residuals act as independent noise driving a linear network. The second derives a self-consistent matrix equation for the covariance matrix. A naive Gaussian closure for the joint statistics of local fields at distinct sites misses cross terms that, in a linear network, would be generated by an external drive. The cavity method recovers these terms from non-Gaussian contributions, revealing an emergent external drive. Higher-order cross-site moments follow a Wick-like decomposition into products of pairwise covariances at leading order, reducing them to the linear-equivalent form. We verify the predictions in simulations. These results extend linear equivalence from feedforward high-dimensional nonlinear systems, where the activations are independent of the weights, to recurrent networks, where the activations are correlated with the couplings that generate them.

We present SplattingAvatar, a hybrid 3D representation of photorealistic human avatars with Gaussian Splatting embedded on a triangle mesh, which renders over 300 FPS on a modern GPU and 30 FPS on a mobile device. We disentangle the motion and appearance of a virtual human with explicit mesh geometry and implicit appearance modeling with Gaussian Splatting. The Gaussians are defined by barycentric coordinates and displacement on a triangle mesh as Phong surfaces. We extend lifted optimization to simultaneously optimize the parameters of the Gaussians while walking on the triangle mesh. SplattingAvatar is a hybrid representation of virtual humans where the mesh represents low-frequency motion and surface deformation, while the Gaussians take over the high-frequency geometry and detailed appearance. Unlike existing deformation methods that rely on an MLP-based linear blend skinning (LBS) field for motion, we control the rotation and translation of the Gaussians directly by mesh, which empowers its compatibility with various animation techniques, e.g., skeletal animation, blend shapes, and mesh editing. Trainable from monocular videos for both full-body and head avatars, SplattingAvatar shows state-of-the-art rendering quality across multiple datasets.

Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible in closed form, so prior methods resort to imprecise approximations of the score matching training objective that degrade performance and preclude applications in high dimensions. In this work, we reexamine these approximations and propose several practical improvements. Our key observation is that most relevant manifolds are symmetric spaces, which are much more amenable to computation. By leveraging and combining various ans\"{a}tze, we can quickly compute relevant quantities to high precision. On low dimensional datasets, our correction produces a noticeable improvement, allowing diffusion to compete with other methods. Additionally, we show that our method enables us to scale to high dimensional tasks on nontrivial manifolds. In particular, we model QCD densities on SU(n) lattices and contrastively learned embeddings on high dimensional hyperspheres.

Gaussian processes (GP) are a well studied Bayesian approach for the optimization of black-box functions. Despite their effectiveness in simple problems, GP-based algorithms hardly scale to high-dimensional functions, as their per-iteration time and space cost is at least quadratic in the number of dimensions d and iterations t. Given a set of A alternatives to choose from, the overall runtime O(t^3A) is prohibitive. In this paper we introduce BKB (budgeted kernelized bandit), a new approximate GP algorithm for optimization under bandit feedback that achieves near-optimal regret (and hence near-optimal convergence rate) with near-constant per-iteration complexity and remarkably no assumption on the input space or covariance of the GP. We combine a kernelized linear bandit algorithm (GP-UCB) with randomized matrix sketching based on leverage score sampling, and we prove that randomly sampling inducing points based on their posterior variance gives an accurate low-rank approximation of the GP, preserving variance estimates and confidence intervals. As a consequence, BKB does not suffer from variance starvation, an important problem faced by many previous sparse GP approximations. Moreover, we show that our procedure selects at most O(d_{eff}) points, where d_{eff} is the effective dimension of the explored space, which is typically much smaller than both d and t. This greatly reduces the dimensionality of the problem, thus leading to a O(TAd_{eff}^2) runtime and O(A d_{eff}) space complexity.

X-ray is widely applied for transmission imaging due to its stronger penetration than natural light. When rendering novel view X-ray projections, existing methods mainly based on NeRF suffer from long training time and slow inference speed. In this paper, we propose a 3D Gaussian splatting-based framework, namely X-Gaussian, for X-ray novel view synthesis. Firstly, we redesign a radiative Gaussian point cloud model inspired by the isotropic nature of X-ray imaging. Our model excludes the influence of view direction when learning to predict the radiation intensity of 3D points. Based on this model, we develop a Differentiable Radiative Rasterization (DRR) with CUDA implementation. Secondly, we customize an Angle-pose Cuboid Uniform Initialization (ACUI) strategy that directly uses the parameters of the X-ray scanner to compute the camera information and then uniformly samples point positions within a cuboid enclosing the scanned object. Experiments show that our X-Gaussian outperforms state-of-the-art methods by 6.5 dB while enjoying less than 15% training time and over 73x inference speed. The application on sparse-view CT reconstruction also reveals the practical values of our method. Code and models will be publicly available at https://github.com/caiyuanhao1998/X-Gaussian . A video demo of the training process visualization is at https://www.youtube.com/watch?v=gDVf_Ngeghg .

Estimating physical properties for visual data is a crucial task in computer vision, graphics, and robotics, underpinning applications such as augmented reality, physical simulation, and robotic grasping. However, this area remains under-explored due to the inherent ambiguities in physical property estimation. To address these challenges, we introduce GaussianProperty, a training-free framework that assigns physical properties of materials to 3D Gaussians. Specifically, we integrate the segmentation capability of SAM with the recognition capability of GPT-4V(ision) to formulate a global-local physical property reasoning module for 2D images. Then we project the physical properties from multi-view 2D images to 3D Gaussians using a voting strategy. We demonstrate that 3D Gaussians with physical property annotations enable applications in physics-based dynamic simulation and robotic grasping. For physics-based dynamic simulation, we leverage the Material Point Method (MPM) for realistic dynamic simulation. For robot grasping, we develop a grasping force prediction strategy that estimates a safe force range required for object grasping based on the estimated physical properties. Extensive experiments on material segmentation, physics-based dynamic simulation, and robotic grasping validate the effectiveness of our proposed method, highlighting its crucial role in understanding physical properties from visual data. Online demo, code, more cases and annotated datasets are available on https://Gaussian-Property.github.io{this https URL}.

Motivated by variational inference methods, we propose a zeroth-order algorithm for solving optimization problems in the space of Gaussian probability measures. The algorithm is based on an interacting system of Gaussian particles that stochastically explore the search space and self-organize around global minima via a consensus-based optimization (CBO) mechanism. Its construction relies on the Linearized Bures-Wasserstein (LBW) space, a novel parametrization of Gaussian measures we introduce for efficient computations. LBW is inspired by linearized optimal transport and preserves key geometric features while enabling computational tractability. We establish well-posedness and study the convergence properties of the particle dynamics via a mean-field approximation. Numerical experiments on variational inference tasks demonstrate the algorithm's robustness and superior performance with respect to deterministic gradient-based method in presence of low-dimensional non log-concave targets.

Despite much progress, achieving real-time high-fidelity head avatar animation is still difficult and existing methods have to trade-off between speed and quality. 3DMM based methods often fail to model non-facial structures such as eyeglasses and hairstyles, while neural implicit models suffer from deformation inflexibility and rendering inefficiency. Although 3D Gaussian has been demonstrated to possess promising capability for geometry representation and radiance field reconstruction, applying 3D Gaussian in head avatar creation remains a major challenge since it is difficult for 3D Gaussian to model the head shape variations caused by changing poses and expressions. In this paper, we introduce PSAvatar, a novel framework for animatable head avatar creation that utilizes discrete geometric primitive to create a parametric morphable shape model and employs 3D Gaussian for fine detail representation and high fidelity rendering. The parametric morphable shape model is a Point-based Morphable Shape Model (PMSM) which uses points instead of meshes for 3D representation to achieve enhanced representation flexibility. The PMSM first converts the FLAME mesh to points by sampling on the surfaces as well as off the meshes to enable the reconstruction of not only surface-like structures but also complex geometries such as eyeglasses and hairstyles. By aligning these points with the head shape in an analysis-by-synthesis manner, the PMSM makes it possible to utilize 3D Gaussian for fine detail representation and appearance modeling, thus enabling the creation of high-fidelity avatars. We show that PSAvatar can reconstruct high-fidelity head avatars of a variety of subjects and the avatars can be animated in real-time (ge 25 fps at a resolution of 512 times 512 ).

Neural rendering techniques have made substantial progress in generating photo-realistic 3D scenes. The latest 3D Gaussian Splatting technique has achieved high quality novel view synthesis as well as fast rendering speed. However, 3D Gaussians lack proficiency in defining accurate 3D geometric structures despite their explicit primitive representations. This is due to the fact that Gaussian's attributes are primarily tailored and fine-tuned for rendering diverse 2D images by their anisotropic nature. To pave the way for efficient 3D reconstruction, we present Spherical Gaussians, a simple and effective representation for 3D geometric boundaries, from which we can directly reconstruct 3D feature curves from a set of calibrated multi-view images. Spherical Gaussians is optimized from grid initialization with a view-based rendering loss, where a 2D edge map is rendered at a specific view and then compared to the ground-truth edge map extracted from the corresponding image, without the need for any 3D guidance or supervision. Given Spherical Gaussians serve as intermedia for the robust edge representation, we further introduce a novel optimization-based algorithm called SGCR to directly extract accurate parametric curves from aligned Spherical Gaussians. We demonstrate that SGCR outperforms existing state-of-the-art methods in 3D edge reconstruction while enjoying great efficiency.

We study a dynamical optimal transport problem on path-space with kinetic cost and nonlocal interaction. For a Gaussian interaction kernel, cyclical monotonicity and the regularity of the effective endpoint cost yield a conservative vector field associated with the initial momentum of the optimal trajectories. Separately, the Gaussian interaction potential W_{π_0} satisfies a uniform gradient estimate and, under a quantitative condition on the interaction strength, is a classical and hence viscosity subsolution of an eikonal equation. We also derive the Euler--Lagrange dynamics directly from path-space optimality by means of endpoint-preserving perturbations of the optimal path measure. The associated phase-space marginals satisfy a measure-valued Liouville-type, or nonlocal kinetic, equation driven by the smooth Gaussian self-consistent force. Conversely, a superposition principle lifts suitable measure-valued solutions of the phase-space continuity equation to measures on phase-space trajectories. Thus path-space optimal transport provides a variational connection between endpoint geometry, eikonal behavior of the Gaussian interaction potential, and kinetic mean-field dynamics.

While 3D Gaussian Splatting has recently become popular for neural rendering, current methods rely on carefully engineered cloning and splitting strategies for placing Gaussians, which can lead to poor-quality renderings, and reliance on a good initialization. In this work, we rethink the set of 3D Gaussians as a random sample drawn from an underlying probability distribution describing the physical representation of the scene-in other words, Markov Chain Monte Carlo (MCMC) samples. Under this view, we show that the 3D Gaussian updates can be converted as Stochastic Gradient Langevin Dynamics (SGLD) updates by simply introducing noise. We then rewrite the densification and pruning strategies in 3D Gaussian Splatting as simply a deterministic state transition of MCMC samples, removing these heuristics from the framework. To do so, we revise the 'cloning' of Gaussians into a relocalization scheme that approximately preserves sample probability. To encourage efficient use of Gaussians, we introduce a regularizer that promotes the removal of unused Gaussians. On various standard evaluation scenes, we show that our method provides improved rendering quality, easy control over the number of Gaussians, and robustness to initialization.

Neural rendering methods have significantly advanced photo-realistic 3D scene rendering in various academic and industrial applications. The recent 3D Gaussian Splatting method has achieved the state-of-the-art rendering quality and speed combining the benefits of both primitive-based representations and volumetric representations. However, it often leads to heavily redundant Gaussians that try to fit every training view, neglecting the underlying scene geometry. Consequently, the resulting model becomes less robust to significant view changes, texture-less area and lighting effects. We introduce Scaffold-GS, which uses anchor points to distribute local 3D Gaussians, and predicts their attributes on-the-fly based on viewing direction and distance within the view frustum. Anchor growing and pruning strategies are developed based on the importance of neural Gaussians to reliably improve the scene coverage. We show that our method effectively reduces redundant Gaussians while delivering high-quality rendering. We also demonstrates an enhanced capability to accommodate scenes with varying levels-of-detail and view-dependent observations, without sacrificing the rendering speed.

Bird's-eye view (BEV) perception has gained significant attention because it provides a unified representation to fuse multiple view images and enables a wide range of down-stream autonomous driving tasks, such as forecasting and planning. Recent state-of-the-art models utilize projection-based methods which formulate BEV perception as query learning to bypass explicit depth estimation. While we observe promising advancements in this paradigm, they still fall short of real-world applications because of the lack of uncertainty modeling and expensive computational requirement. In this work, we introduce GaussianLSS, a novel uncertainty-aware BEV perception framework that revisits unprojection-based methods, specifically the Lift-Splat-Shoot (LSS) paradigm, and enhances them with depth un-certainty modeling. GaussianLSS represents spatial dispersion by learning a soft depth mean and computing the variance of the depth distribution, which implicitly captures object extents. We then transform the depth distribution into 3D Gaussians and rasterize them to construct uncertainty-aware BEV features. We evaluate GaussianLSS on the nuScenes dataset, achieving state-of-the-art performance compared to unprojection-based methods. In particular, it provides significant advantages in speed, running 2.5x faster, and in memory efficiency, using 0.3x less memory compared to projection-based methods, while achieving competitive performance with only a 0.4% IoU difference.

Recent successful generative models are trained by fitting a neural network to an a-priori defined tractable probability density path taking noise to training examples. In this paper we investigate the space of Gaussian probability paths, which includes diffusion paths as an instance, and look for an optimal member in some useful sense. In particular, minimizing the Kinetic Energy (KE) of a path is known to make particles' trajectories simple, hence easier to sample, and empirically improve performance in terms of likelihood of unseen data and sample generation quality. We investigate Kinetic Optimal (KO) Gaussian paths and offer the following observations: (i) We show the KE takes a simplified form on the space of Gaussian paths, where the data is incorporated only through a single, one dimensional scalar function, called the data separation function. (ii) We characterize the KO solutions with a one dimensional ODE. (iii) We approximate data-dependent KO paths by approximating the data separation function and minimizing the KE. (iv) We prove that the data separation function converges to 1 in the general case of arbitrary normalized dataset consisting of n samples in d dimension as n/drightarrow 0. A consequence of this result is that the Conditional Optimal Transport (Cond-OT) path becomes kinetic optimal as n/drightarrow 0. We further support this theory with empirical experiments on ImageNet.

In this work we propose an analytical model that reproduces the core-bounds phase of gravitational waves (GW) of Rapidly Rotating (RR) from Core Collapse Supernovae (CCSNe), as a function of three parameters, the arrival time tau, the ratio of the kinetic and potential energy beta and a phenomenological parameter alpha related to rotation and equation of state (EOS). To validate the model we use 126 waveforms from the Richers catalog Richers_2017 selected with the criteria of exploring a range of rotation profiles, and involving EOS. To quantify the degree of accuracy of the proposed model, with a particular focus on the rotation parameter beta, we show that the average Fitting Factor (FF) between the simulated waveforms with the templates is 94.4\%. In order to estimate the parameters we propose a frequentist matched filtering approach in real interferometric noise which does not require assigning any priors. We use the Matched Filter (MF) technique, where we inject a bank of templates considering simulated colored Gaussian noise and the real noise of O3L1. For example for A300w6.00\_BHBLP at 10Kpc we obtain a standar deviation of sigma = 3.34times 10^{-3} for simulated colored Gaussian noise and sigma= 1.46times 10^{-2} for real noise. On the other hand, from the asymptotic expansion of the variance we obtain the theoretical minimum error for beta at 10 kpc and optimal orientation. The estimation error in this case is from 10^{-2} to 10^{-3} as beta increases. We show that the results of the estimation error of beta for the 3-parameter space (3D) is consistent with the single-parameter space (1D), which allows us to conclude that beta is decoupled from the others two parameters.

We revisit the flat-sky approximation for evaluating the angular power spectra of projected random fields by retaining information about the correlations along the line of sight. With broad, overlapping radial window functions, these line-of-sight correlations are suppressed and are ignored in the Limber approximation. However, retaining the correlations is important for narrow window functions or unequal-time spectra but introduces significant computational difficulties due to the highly oscillatory nature of the integrands involved. We deal with the integral over line-of-sight wave-modes in the flat-sky approximation analytically, using the FFTlog expansion of the 3D power spectrum. This results in an efficient computational method, which is a substantial improvement compared to any full-sky approaches. We apply our results to galaxy clustering (with and without redshift-space distortions), CMB lensing and galaxy lensing observables. For clustering, we find excellent agreement with the full-sky results on large (percent-level agreement) and intermediate or small (subpercent agreement) scales, dramatically out-performing the Limber approximation for both wide and narrow window functions, and in equal- and unequal-time cases. In the case of lensing, we show on the full sky that the angular power spectrum of the convergence can be very well approximated by projecting the 3D Laplacian (rather than the correct angular Laplacian) of the gravitational potential, even on large scales. Combining this approximation with our flat-sky techniques provides an efficient and accurate evaluation of the CMB lensing angular power spectrum on all scales.

We analyse the robustness of the DESI 2024 cosmological inference from fits to the full shape of the galaxy power spectrum to uncertainties in the Halo Occupation Distribution (HOD) model of the galaxy-halo connection and the choice of priors on nuisance parameters. We assess variations in the recovered cosmological parameters across a range of mocks populated with different HOD models and find that shifts are often greater than 20% of the expected statistical uncertainties from the DESI data. We encapsulate the effect of such shifts in terms of a systematic covariance term, C_{rm HOD}, and an additional diagonal contribution quantifying the impact of our choice of nuisance parameter priors on the ability of the effective field theory (EFT) model to correctly recover the cosmological parameters of the simulations. These two covariance contributions are designed to be added to the usual covariance term, C_{rm stat}, describing the statistical uncertainty in the power spectrum measurement, in order to fairly represent these sources of systematic uncertainty. This approach is more general and robust to choices of model free parameters or additional external datasets used in cosmological fits than the alternative approach of adding systematic uncertainties at the level of the recovered marginalised parameter posteriors. We compare the approaches within the context of a fixed LambdaCDM model and demonstrate that our method gives conservative estimates of the systematic uncertainty that nevertheless have little impact on the final posteriors obtained from DESI data.

Reconstructing photo-realistic drivable human avatars from multi-view image sequences has been a popular and challenging topic in the field of computer vision and graphics. While existing NeRF-based methods can achieve high-quality novel view rendering of human models, both training and inference processes are time-consuming. Recent approaches have utilized 3D Gaussians to represent the human body, enabling faster training and rendering. However, they undermine the importance of the mesh guidance and directly predict Gaussians in 3D space with coarse mesh guidance. This hinders the learning procedure of the Gaussians and tends to produce blurry textures. Therefore, we propose UV Gaussians, which models the 3D human body by jointly learning mesh deformations and 2D UV-space Gaussian textures. We utilize the embedding of UV map to learn Gaussian textures in 2D space, leveraging the capabilities of powerful 2D networks to extract features. Additionally, through an independent Mesh network, we optimize pose-dependent geometric deformations, thereby guiding Gaussian rendering and significantly enhancing rendering quality. We collect and process a new dataset of human motion, which includes multi-view images, scanned models, parametric model registration, and corresponding texture maps. Experimental results demonstrate that our method achieves state-of-the-art synthesis of novel view and novel pose. The code and data will be made available on the homepage https://alex-jyj.github.io/UV-Gaussians/ once the paper is accepted.

Amid mounting concern about the reliability and credibility of machine learning research, we present a principled framework for making robust and generalizable claims: the multiverse analysis. Our framework builds upon the multiverse analysis (Steegen et al., 2016) introduced in response to psychology's own reproducibility crisis. To efficiently explore high-dimensional and often continuous ML search spaces, we model the multiverse with a Gaussian Process surrogate and apply Bayesian experimental design. Our framework is designed to facilitate drawing robust scientific conclusions about model performance, and thus our approach focuses on exploration rather than conventional optimization. In the first of two case studies, we investigate disputed claims about the relative merit of adaptive optimizers. Second, we synthesize conflicting research on the effect of learning rate on the large batch training generalization gap. For the machine learning community, the multiverse analysis is a simple and effective technique for identifying robust claims, for increasing transparency, and a step toward improved reproducibility.

Given a Finsler manifold (M,F) and suitable nested regions B_1subset B_2subset B_3, with Msetminus B_1 connected and non-contractible, as in the model case of concentric balls in mathbb R^n, ngeq2, we consider conformal deformations φ_λF which can be chosen so as to leave F unchanged outside B_3 and grow on B_2setminus B_1. This growth creates an energy barrier, diverging with λ, that low-energy curves cannot cross. The minimax families can therefore be chosen in the path space of Msetminus B_1, which contains compact subsets of arbitrarily large category. Consequently, for every minmathbb N and all sufficiently large λ, there exist at least m geodesics joining two prescribed points, all avoiding B_1. This extends to the Finsler setting an idea of Capozzi, Fortunato and Greco. The same scheme applies to Randers metrics α+β, with α deformed and β unchanged, as well as to fixed-energy trajectories of non-relativistic electromagnetic systems and relativistic Lorentz-force trajectories, and light rays in spacetimes endowed with a causal Killing field. In the non-relativistic case only the scalar potential is perturbed, while the magnetic field is preserved. When the Killing field is not everywhere timelike, the Fermat principle yields a possibly singular Randers--Kropina metric on a spacelike hypersurface. Under a natural admissible category condition, combining localized barriers with Randers approximation gives m Randers--Kropina geodesics with pairwise distinct lengths, equivalently m future-pointing light rays with pairwise distinct arrival times.

We construct a geometric framework for cosmological large-scale structure based on optimal transport theory and Wasserstein geometry. In this framework, Ricci curvature on the probability measure space P_2(M) is characterized by the geodesic convexity of entropy and is formulated as the response of probability distributions to optimal transport. We introduce effective Ricci curvatures K_{eff}^{(infty)} and K_{eff}^{(N)} associated with Kullback--Leibler-type and Rényi-type entropies, corresponding respectively to the curvature-dimension conditions CD(K,infty) and CD(K,N). By localizing these curvatures to finite scales using local and reference measures, we construct curvature indicators applicable to observational data. Under a local quadratic approximation, the effective curvature reduces to the Hessian of the log-density, showing that conventional Hessian-based structure classifications arise as a limiting case of the present framework. We further show that effective curvature depends on observational scale and formulate this dependence as a scale flow, distinct from Ricci flow because it describes a change of resolution rather than a time evolution of geometry. Treating curvature as a random field then extends the statistical description of density fields: curvature statistics are given by higher-order weighted integrals of the power spectrum and by spatial derivatives of the correlation function, emphasizing geometric rather than amplitude information. This framework provides a unified connection between optimal transport geometry and cosmological structure analysis, and offers a new perspective on multiscale structure and nonlinear statistics.

Sparse Multi-view Images can be Learned to predict explicit radiance fields via Generalizable Gaussian Splatting approaches, which can achieve wider application prospects in real-life when ground-truth camera parameters are not required as inputs. In this paper, a novel generalizable Gaussian Splatting method, SmileSplat, is proposed to reconstruct pixel-aligned Gaussian surfels for diverse scenarios only requiring unconstrained sparse multi-view images. First, Gaussian surfels are predicted based on the multi-head Gaussian regression decoder, which can are represented with less degree-of-freedom but have better multi-view consistency. Furthermore, the normal vectors of Gaussian surfel are enhanced based on high-quality of normal priors. Second, the Gaussians and camera parameters (both extrinsic and intrinsic) are optimized to obtain high-quality Gaussian radiance fields for novel view synthesis tasks based on the proposed Bundle-Adjusting Gaussian Splatting module. Extensive experiments on novel view rendering and depth map prediction tasks are conducted on public datasets, demonstrating that the proposed method achieves state-of-the-art performance in various 3D vision tasks. More information can be found on our project page (https://yanyan-li.github.io/project/gs/smilesplat)

Four-dimensional computed tomography (4D CT) reconstruction is crucial for capturing dynamic anatomical changes but faces inherent limitations from conventional phase-binning workflows. Current methods discretize temporal resolution into fixed phases with respiratory gating devices, introducing motion misalignment and restricting clinical practicality. In this paper, We propose X^2-Gaussian, a novel framework that enables continuous-time 4D-CT reconstruction by integrating dynamic radiative Gaussian splatting with self-supervised respiratory motion learning. Our approach models anatomical dynamics through a spatiotemporal encoder-decoder architecture that predicts time-varying Gaussian deformations, eliminating phase discretization. To remove dependency on external gating devices, we introduce a physiology-driven periodic consistency loss that learns patient-specific breathing cycles directly from projections via differentiable optimization. Extensive experiments demonstrate state-of-the-art performance, achieving a 9.93 dB PSNR gain over traditional methods and 2.25 dB improvement against prior Gaussian splatting techniques. By unifying continuous motion modeling with hardware-free period learning, X^2-Gaussian advances high-fidelity 4D CT reconstruction for dynamic clinical imaging. Code is publicly available at: https://x2-gaussian.github.io/.

We present the measurements and cosmological implications of the galaxy two-point clustering using over 4.7 million unique galaxy and quasar redshifts in the range 0.1<z<2.1 divided into six redshift bins over a sim 7,500 square degree footprint, from the first year of observations with the Dark Energy Spectroscopic Instrument (DESI Data Release 1). By fitting the full power spectrum, we extend previous DESI DR1 baryon acoustic oscillation (BAO) measurements to include redshift-space distortions and signals from the matter-radiation equality scale. For the first time, this Full-Shape analysis is blinded at the catalogue-level to avoid confirmation bias and the systematic errors are accounted for at the two-point clustering level, which automatically propagates them into any cosmological parameter. When analysing the data in terms of compressed model-agnostic variables, we obtain a combined precision of 4.7\% on the amplitude of the redshift space distortion signal reaching similar precision with just one year of DESI data than with 20 years of observation from previous generation surveys. We analyse the data to directly constrain the cosmological parameters within the LambdaCDM model using perturbation theory and combine this information with the reconstructed DESI DR1 galaxy BAO. Using a Big Bang Nucleosynthesis Gaussian prior on the baryon density parameter, and a Gaussian prior on the spectral index, we constrain the matter density is Omega_m=0.296pm 0.010 and the Hubble constant H_0=(68.63 pm 0.79)[{rm km, s^{-1}Mpc^{-1}}]. Additionally, we measure the amplitude of clustering sigma_8=0.841 pm 0.034. The DESI DR1 results are in agreement with the LambdaCDM model based on general relativity with parameters consistent with those from Planck. The cosmological interpretation of these results in combination with external datasets are presented in a companion paper.

We present the first linear polarization measurements from the 2015 long-duration balloon flight of SPIDER, an experiment designed to map the polarization of the cosmic microwave background (CMB) on degree angular scales. Results from these measurements include maps and angular power spectra from observations of 4.8% of the sky at 95 and 150 GHz, along with the results of internal consistency tests on these data. While the polarized CMB anisotropy from primordial density perturbations is the dominant signal in this region of sky, Galactic dust emission is also detected with high significance; Galactic synchrotron emission is found to be negligible in the SPIDER bands. We employ two independent foreground-removal techniques in order to explore the sensitivity of the cosmological result to the assumptions made by each. The primary method uses a dust template derived from Planck data to subtract the Galactic dust signal. A second approach, employing a joint analysis of SPIDER and Planck data in the harmonic domain, assumes a modified-blackbody model for the spectral energy distribution of the dust with no constraint on its spatial morphology. Using a likelihood that jointly samples the template amplitude and r parameter space, we derive 95% upper limits on the primordial tensor-to-scalar ratio from Feldman-Cousins and Bayesian constructions, finding r<0.11 and r<0.19, respectively. Roughly half the uncertainty in r derives from noise associated with the template subtraction. New data at 280 GHz from SPIDER's second flight will complement the Planck polarization maps, providing powerful measurements of the polarized Galactic dust emission.

Modeling animatable human avatars from RGB videos is a long-standing and challenging problem. Recent works usually adopt MLP-based neural radiance fields (NeRF) to represent 3D humans, but it remains difficult for pure MLPs to regress pose-dependent garment details. To this end, we introduce Animatable Gaussians, a new avatar representation that leverages powerful 2D CNNs and 3D Gaussian splatting to create high-fidelity avatars. To associate 3D Gaussians with the animatable avatar, we learn a parametric template from the input videos, and then parameterize the template on two front \& back canonical Gaussian maps where each pixel represents a 3D Gaussian. The learned template is adaptive to the wearing garments for modeling looser clothes like dresses. Such template-guided 2D parameterization enables us to employ a powerful StyleGAN-based CNN to learn the pose-dependent Gaussian maps for modeling detailed dynamic appearances. Furthermore, we introduce a pose projection strategy for better generalization given novel poses. Overall, our method can create lifelike avatars with dynamic, realistic and generalized appearances. Experiments show that our method outperforms other state-of-the-art approaches. Code: https://github.com/lizhe00/AnimatableGaussians

Simulation-based inference with conditional neural density estimators is a powerful approach to solving inverse problems in science. However, these methods typically treat the underlying forward model as a black box, with no way to exploit geometric properties such as equivariances. Equivariances are common in scientific models, however integrating them directly into expressive inference networks (such as normalizing flows) is not straightforward. We here describe an alternative method to incorporate equivariances under joint transformations of parameters and data. Our method -- called group equivariant neural posterior estimation (GNPE) -- is based on self-consistently standardizing the "pose" of the data while estimating the posterior over parameters. It is architecture-independent, and applies both to exact and approximate equivariances. As a real-world application, we use GNPE for amortized inference of astrophysical binary black hole systems from gravitational-wave observations. We show that GNPE achieves state-of-the-art accuracy while reducing inference times by three orders of magnitude.

A mixed Gaussian fractional process {Y(t)}_{t Bbb R} = {PX(t)}_{t Bbb R} is a multivariate stochastic process obtained by pre-multiplying a vector of independent, Gaussian fractional process entries X by a nonsingular matrix P. It is interpreted that Y is observable, while X is a hidden process occurring in an (unknown) system of coordinates P. Mixed processes naturally arise as approximations to solutions of physically relevant classes of multivariate fractional SDEs under aggregation. We propose a semiparametric two-step wavelet-based method for estimating both the demixing matrix P^{-1} and the memory parameters of X. The asymptotic normality of the estimators is established both in continuous and discrete time. Monte Carlo experiments show that the finite sample estimation performance is comparable to that of parametric methods, while being very computationally efficient. As applications, we model a bivariate time series of annual tree ring width measurements, and establish the asymptotic normality of the eigenstructure of sample wavelet matrices.

We present a method that simultaneously addresses the tasks of dynamic scene novel-view synthesis and six degree-of-freedom (6-DOF) tracking of all dense scene elements. We follow an analysis-by-synthesis framework, inspired by recent work that models scenes as a collection of 3D Gaussians which are optimized to reconstruct input images via differentiable rendering. To model dynamic scenes, we allow Gaussians to move and rotate over time while enforcing that they have persistent color, opacity, and size. By regularizing Gaussians' motion and rotation with local-rigidity constraints, we show that our Dynamic 3D Gaussians correctly model the same area of physical space over time, including the rotation of that space. Dense 6-DOF tracking and dynamic reconstruction emerges naturally from persistent dynamic view synthesis, without requiring any correspondence or flow as input. We demonstrate a large number of downstream applications enabled by our representation, including first-person view synthesis, dynamic compositional scene synthesis, and 4D video editing.

3D Gaussian Splatting is a new method for modeling and rendering 3D radiance fields that achieves much faster learning and rendering time compared to SOTA NeRF methods. However, it comes with a drawback in the much larger storage demand compared to NeRF methods since it needs to store the parameters for several 3D Gaussians. We notice that many Gaussians may share similar parameters, so we introduce a simple vector quantization method based on \kmeans algorithm to quantize the Gaussian parameters. Then, we store the small codebook along with the index of the code for each Gaussian. Moreover, we compress the indices further by sorting them and using a method similar to run-length encoding. We do extensive experiments on standard benchmarks as well as a new benchmark which is an order of magnitude larger than the standard benchmarks. We show that our simple yet effective method can reduce the storage cost for the original 3D Gaussian Splatting method by a factor of almost 20times with a very small drop in the quality of rendered images.

The approximation of Partial Differential Equations (PDEs) using neural networks has seen significant advancements through Physics-Informed Neural Networks (PINNs). Despite their straightforward optimization framework and flexibility in implementing various PDEs, PINNs often suffer from limited accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which struggle to effectively learn high-frequency and non-linear components. Recently, parametric mesh representations in combination with neural networks have been investigated as a promising approach to eliminate the inductive biases of neural networks. However, they usually require very high-resolution grids and a large number of collocation points to achieve high accuracy while avoiding overfitting issues. In addition, the fixed positions of the mesh parameters restrict their flexibility, making it challenging to accurately approximate complex PDEs. To overcome these limitations, we propose Physics-Informed Gaussians (PIGs), which combine feature embeddings using Gaussian functions with a lightweight neural network. Our approach uses trainable parameters for the mean and variance of each Gaussian, allowing for dynamic adjustment of their positions and shapes during training. This adaptability enables our model to optimally approximate PDE solutions, unlike models with fixed parameter positions. Furthermore, the proposed approach maintains the same optimization framework used in PINNs, allowing us to benefit from their excellent properties. Experimental results show the competitive performance of our model across various PDEs, demonstrating its potential as a robust tool for solving complex PDEs. Our project page is available at https://namgyukang.github.io/Physics-Informed-Gaussians/

We present a method for joint inference of cosmological parameters and convergence maps from weak lensing observations, targeting the full posterior conditioned on the observed shear field. Our approach uses implicit inference with diffusion models, learning the joint distribution from simulations, without the need to have an explicit and differentiable forward model for gradient-based MCMC sampling. We introduce a transformer-based architecture that operates in pixel space and treats cosmological parameters as additional tokens in a unified sequence, enabling efficient multimodal processing within a single network. At inference time, the trained model generates posterior samples of joint convergence maps and cosmological parameters conditioned on observed noisy shear fields. We demonstrate the method on simulated weak lensing data generated from log-normal fields in a wcdm cosmology. The model accurately reconstructs convergence maps and recovers cosmological posteriors that agree with traditional MCMC, while remaining well calibrated across the prior, with a MIRA calibration score of 0.635 pm 0.017 on the joint posterior (where 0.667 is optimal). The inferred fields reproduce the correct two-point statistics as well as non-Gaussian statistics such as the one-point distribution. This work establishes diffusion-based implicit inference as a viable route toward full field-level cosmological analyses, paving the way for applications to more realistic, non-differentiable simulators.

Despite their many desirable properties, Gaussian processes (GPs) are often compared unfavorably to deep neural networks (NNs) for lacking the ability to learn representations. Recent efforts to bridge the gap between GPs and deep NNs have yielded a new class of inter-domain variational GPs in which the inducing variables correspond to hidden units of a feedforward NN. In this work, we examine some practical issues associated with this approach and propose an extension that leverages the orthogonal decomposition of GPs to mitigate these limitations. In particular, we introduce spherical inter-domain features to construct more flexible data-dependent basis functions for both the principal and orthogonal components of the GP approximation and show that incorporating NN activation features under this framework not only alleviates these shortcomings but is more scalable than alternative strategies. Experiments on multiple benchmark datasets demonstrate the effectiveness of our approach.

Recent Pulsar Timing Array datasets provide compelling evidence for a nano-Hertz gravitational-wave background, but robust detection requires characterizing statistical fluctuations of the Hellings-Downs (HD) correlation expected from a finite population of discrete sources. Building on the variance calculation of Allen (2023), we derive the third central moment (skewness) of the HD correlation for a single unpolarized point source and an ensemble of many interfering point sources in the confusion-noise regime. To isolate the intrinsic non-Gaussianity of the background, we extend the pulsar-averaging formalism to third order by introducing a three-point averaged correlation function, which allows us to define the cosmic skewness. We find that the skewness remains non-zero in the large-source-number limit and is controlled by a new geometric three-point function. These results suggest that incorporating higher-order moments could provide additional information on source discreteness beyond standard Gaussian analyses.

We develop a novel way to probe subgalactic-scale matter distribution with diffractive lensing on gravitational waves. Five-year observations from Einstein Telescope and DECIGO are expected to probe k= 10^5sim 10^8 ,{rm Mpc}^{-1} down to P(k) = 10^{-16} sim 10^{-14} ,{rm Mpc}^3 level. These results can be interpreted in terms of primordial black holes in the range M_{rm PBH} gtrsim 10^{-3}M_odot down to f_{rm PBH} = 10^{-6} level, or QCD axion minihalos in the range m_a = 10^{-3} sim 10^{-12} ,{rm eV}. A key result of the paper is the approximate relation between the scale k and the gravitational wave frequency f, derived in an ensemble of `multi-lensing' events. This relation enables direct measurement of the power spectrum at specific scales, with sensitivities characterized by model-independent kernels delta P(k). Additionally, we delineate the statistical properties of `multi-lensing' based on the `Fresnel number' N_F. When N_F cal O(1), the statistical significance can be approximately calculated by Variance of lensing effects, which is directly related to the power spectrum among other moments of matter distribution.

Current learning-based methods predict NeRF or 3D Gaussians from point clouds to achieve photo-realistic rendering but still depend on categorical priors, dense point clouds, or additional refinements. Hence, we introduce a novel point cloud rendering method by predicting 2D Gaussians from point clouds. Our method incorporates two identical modules with an entire-patch architecture enabling the network to be generalized to multiple datasets. The module normalizes and initializes the Gaussians utilizing the point cloud information including normals, colors and distances. Then, splitting decoders are employed to refine the initial Gaussians by duplicating them and predicting more accurate results, making our methodology effectively accommodate sparse point clouds as well. Once trained, our approach exhibits direct generalization to point clouds across different categories. The predicted Gaussians are employed directly for rendering without additional refinement on the rendered images, retaining the benefits of 2D Gaussians. We conduct extensive experiments on various datasets, and the results demonstrate the superiority and generalization of our method, which achieves SOTA performance. The code is available at https://github.com/murcherful/GauPCRender}{https://github.com/murcherful/GauPCRender.

Sequential VAEs have been successfully considered for many high-dimensional time series modelling problems, with many variant models relying on discrete-time mechanisms such as recurrent neural networks (RNNs). On the other hand, continuous-time methods have recently gained attraction, especially in the context of irregularly-sampled time series, where they can better handle the data than discrete-time methods. One such class are Gaussian process variational autoencoders (GPVAEs), where the VAE prior is set as a Gaussian process (GP). However, a major limitation of GPVAEs is that it inherits the cubic computational cost as GPs, making it unattractive to practioners. In this work, we leverage the equivalent discrete state space representation of Markovian GPs to enable linear time GPVAE training via Kalman filtering and smoothing. We show on a variety of high-dimensional temporal and spatiotemporal tasks that our method performs favourably compared to existing approaches whilst being computationally highly scalable.

Time series modeling is a well-established problem, which often requires that methods (1) expressively represent complicated dependencies, (2) forecast long horizons, and (3) efficiently train over long sequences. State-space models (SSMs) are classical models for time series, and prior works combine SSMs with deep learning layers for efficient sequence modeling. However, we find fundamental limitations with these prior approaches, proving their SSM representations cannot express autoregressive time series processes. We thus introduce SpaceTime, a new state-space time series architecture that improves all three criteria. For expressivity, we propose a new SSM parameterization based on the companion matrix -- a canonical representation for discrete-time processes -- which enables SpaceTime's SSM layers to learn desirable autoregressive processes. For long horizon forecasting, we introduce a "closed-loop" variation of the companion SSM, which enables SpaceTime to predict many future time-steps by generating its own layer-wise inputs. For efficient training and inference, we introduce an algorithm that reduces the memory and compute of a forward pass with the companion matrix. With sequence length ell and state-space size d, we go from O(d ell) na\"ively to O(d + ell). In experiments, our contributions lead to state-of-the-art results on extensive and diverse benchmarks, with best or second-best AUROC on 6 / 7 ECG and speech time series classification, and best MSE on 14 / 16 Informer forecasting tasks. Furthermore, we find SpaceTime (1) fits AR(p) processes that prior deep SSMs fail on, (2) forecasts notably more accurately on longer horizons than prior state-of-the-art, and (3) speeds up training on real-world ETTh1 data by 73% and 80% relative wall-clock time over Transformers and LSTMs.

Belief Propagation (BP) is a powerful algorithm for distributed inference in probabilistic graphical models, however it quickly becomes infeasible for practical compute and memory budgets. Many efficient, non-parametric forms of BP have been developed, but the most popular is Gaussian Belief Propagation (GBP), a variant that assumes all distributions are locally Gaussian. GBP is widely used due to its efficiency and empirically strong performance in applications like computer vision or sensor networks - even when modelling non-Gaussian problems. In this paper, we seek to provide a theoretical guarantee for when Gaussian approximations are valid in highly non-Gaussian, sparsely-connected factor graphs performing BP (common in spatial AI). We leverage the Central Limit Theorem (CLT) to prove mathematically that variables' beliefs under BP converge to a Gaussian distribution in complex, loopy factor graphs obeying our 4 key assumptions. We then confirm experimentally that variable beliefs become increasingly Gaussian after just a few BP iterations in a stereo depth estimation task.

In the context of C^* dynamical systems, we consider a locally compact group G acting by ^*-automorphisms on a C^* algebra U of observables, and assume a state of U that satisfies the clustering property with respect to a net of group elements of G. That is, the two-point connected correlation function vanishes in the limit on the net, when one observable is translated under the group action. Then we show that all higher order connected correlation functions (Ursell functions, or classical cumulants) and all free correlation functions (free cumulants, from free probability) vanish at the same rate in that limit. Additionally, we show that mean clustering, also called ergodicity, extends to higher order correlations. We then apply those results to equilibrium states of quantum spin lattice models. Under certain assumptions on the range of the interaction, high temperature Gibbs states are known to be exponentially clustering w.r.t. space translations. Combined with the Lieb-Robinson bound, one obtains exponential clustering for space-time translations outside the Lieb-Robinson light-cone. Therefore, by our present results, all the higher order connected and free correlation functions will vanish exponentially under space-time translations outside the Lieb-Robinson light cone, in high temperature Gibbs states. Another consequence is that their long-time averaging over a space-time ray vanishes for almost every ray velocity.

Gaussian analysis of new, high-angular-resolution interstellar 21-cm neutral hydrogen emission profile structure more clearly reveals the presence of the previously reported signature of the critical ionization velocity ({\it CIV}) of Helium (34 km s^{-1}). The present analysis includes 1496 component line widths for 178 neutral hydrogen profiles in two areas of sky at galactic latitudes around -50^circ, well away from the galactic plane. The new data considered here allow the interstellar abundance of Helium to be calculated, and the derived value of 0.095 pm 0.020 compares extremely well with the value of 0.085 for the cosmic abundance based on solar data. Although the precise mechanisms that give rise to the {\it CIV} effect in interstellar space are not yet understood, our results may provide additional motivation for further theoretical study of how the mechanism operates.

We develop a unified mathematical framework for certified Top-k attention truncation that quantifies approximation error at both the distribution and output levels. For a single attention distribution P and its Top-k truncation hat P, we show that the total-variation distance coincides with the discarded softmax tail mass and satisfies TV(P,hat P)=1-e^{-KL(hat PVert P)}, yielding sharp Top-k-specific bounds in place of generic inequalities. From this we derive non-asymptotic deterministic bounds -- from a single boundary gap through multi-gap and blockwise variants -- that control TV(P,hat P) using only the ordered logits. Using an exact head-tail decomposition, we prove that the output error factorizes as |Attn(q,K,V)-Attn_k(q,K,V)|_2=τ|μ_{tail}-μ_{head}|_2 with τ=TV(P,hat P), yielding a new head-tail diameter bound |Attn(q,K,V)-Attn_k(q,K,V)|_2leτ,diam_{H,T} and refinements linking the error to Var_P(V). Under an i.i.d. Gaussian score model s_isimmathcal N(μ,σ^2) we derive closed-form tail masses and an asymptotic rule for the minimal k_varepsilon ensuring TV(P,hat P)levarepsilon, namely k_varepsilon/napproxΦ_c(σ+Φ^{-1}(varepsilon)). Experiments on bert-base-uncased and synthetic logits confirm the predicted scaling of k_varepsilon/n and show that certified Top-k can reduce scored keys by 2-4times on average while meeting the prescribed total-variation budget.

In this paper, we introduce GS-LIVM, a real-time photo-realistic LiDAR-Inertial-Visual mapping framework with Gaussian Splatting tailored for outdoor scenes. Compared to existing methods based on Neural Radiance Fields (NeRF) and 3D Gaussian Splatting (3DGS), our approach enables real-time photo-realistic mapping while ensuring high-quality image rendering in large-scale unbounded outdoor environments. In this work, Gaussian Process Regression (GPR) is employed to mitigate the issues resulting from sparse and unevenly distributed LiDAR observations. The voxel-based 3D Gaussians map representation facilitates real-time dense mapping in large outdoor environments with acceleration governed by custom CUDA kernels. Moreover, the overall framework is designed in a covariance-centered manner, where the estimated covariance is used to initialize the scale and rotation of 3D Gaussians, as well as update the parameters of the GPR. We evaluate our algorithm on several outdoor datasets, and the results demonstrate that our method achieves state-of-the-art performance in terms of mapping efficiency and rendering quality. The source code is available on GitHub.

Approximate inference in Gaussian process (GP) models with non-conjugate likelihoods gets entangled with the learning of the model hyperparameters. We improve hyperparameter learning in GP models and focus on the interplay between variational inference (VI) and the learning target. While VI's lower bound to the marginal likelihood is a suitable objective for inferring the approximate posterior, we show that a direct approximation of the marginal likelihood as in Expectation Propagation (EP) is a better learning objective for hyperparameter optimization. We design a hybrid training procedure to bring the best of both worlds: it leverages conjugate-computation VI for inference and uses an EP-like marginal likelihood approximation for hyperparameter learning. We compare VI, EP, Laplace approximation, and our proposed training procedure and empirically demonstrate the effectiveness of our proposal across a wide range of data sets.

This paper studies Kernel density estimation for a high-dimensional distribution rho(x). Traditional approaches have focused on the limit of large number of data points n and fixed dimension d. We analyze instead the regime where both the number n of data points y_i and their dimensionality d grow with a fixed ratio alpha=(log n)/d. Our study reveals three distinct statistical regimes for the kernel-based estimate of the density hat rho_h^{D}(x)=1{n h^d}sum_{i=1}^n Kleft(x-y_i{h}right), depending on the bandwidth h: a classical regime for large bandwidth where the Central Limit Theorem (CLT) holds, which is akin to the one found in traditional approaches. Below a certain value of the bandwidth, h_{CLT}(alpha), we find that the CLT breaks down. The statistics of hat rho_h^{D}(x) for a fixed x drawn from rho(x) is given by a heavy-tailed distribution (an alpha-stable distribution). In particular below a value h_G(alpha), we find that hat rho_h^{D}(x) is governed by extreme value statistics: only a few points in the database matter and give the dominant contribution to the density estimator. We provide a detailed analysis for high-dimensional multivariate Gaussian data. We show that the optimal bandwidth threshold based on Kullback-Leibler divergence lies in the new statistical regime identified in this paper. Our findings reveal limitations of classical approaches, show the relevance of these new statistical regimes, and offer new insights for Kernel density estimation in high-dimensional settings.

Constructing a 3D scene capable of accommodating open-ended language queries, is a pivotal pursuit, particularly within the domain of robotics. Such technology facilitates robots in executing object manipulations based on human language directives. To tackle this challenge, some research efforts have been dedicated to the development of language-embedded implicit fields. However, implicit fields (e.g. NeRF) encounter limitations due to the necessity of processing a large number of input views for reconstruction, coupled with their inherent inefficiencies in inference. Thus, we present the GaussianGrasper, which utilizes 3D Gaussian Splatting to explicitly represent the scene as a collection of Gaussian primitives. Our approach takes a limited set of RGB-D views and employs a tile-based splatting technique to create a feature field. In particular, we propose an Efficient Feature Distillation (EFD) module that employs contrastive learning to efficiently and accurately distill language embeddings derived from foundational models. With the reconstructed geometry of the Gaussian field, our method enables the pre-trained grasping model to generate collision-free grasp pose candidates. Furthermore, we propose a normal-guided grasp module to select the best grasp pose. Through comprehensive real-world experiments, we demonstrate that GaussianGrasper enables robots to accurately query and grasp objects with language instructions, providing a new solution for language-guided manipulation tasks. Data and codes can be available at https://github.com/MrSecant/GaussianGrasper.

Novel view synthesis has advanced significantly with the development of neural radiance fields (NeRF) and 3D Gaussian splatting (3DGS). However, achieving high quality without compromising real-time rendering remains challenging, particularly for physically-based ray tracing with view-dependent effects. Recently, N-dimensional Gaussians (N-DG) introduced a 6D spatial-angular representation to better incorporate view-dependent effects, but the Gaussian representation and control scheme are sub-optimal. In this paper, we revisit 6D Gaussians and introduce 6D Gaussian Splatting (6DGS), which enhances color and opacity representations and leverages the additional directional information in the 6D space for optimized Gaussian control. Our approach is fully compatible with the 3DGS framework and significantly improves real-time radiance field rendering by better modeling view-dependent effects and fine details. Experiments demonstrate that 6DGS significantly outperforms 3DGS and N-DG, achieving up to a 15.73 dB improvement in PSNR with a reduction of 66.5% Gaussian points compared to 3DGS. The project page is: https://gaozhongpai.github.io/6dgs/

Score-based generative models (SGMs) are a powerful class of generative models that exhibit remarkable empirical performance. Score-based generative modelling (SGM) consists of a ``noising'' stage, whereby a diffusion is used to gradually add Gaussian noise to data, and a generative model, which entails a ``denoising'' process defined by approximating the time-reversal of the diffusion. Existing SGMs assume that data is supported on a Euclidean space, i.e. a manifold with flat geometry. In many domains such as robotics, geoscience or protein modelling, data is often naturally described by distributions living on Riemannian manifolds and current SGM techniques are not appropriate. We introduce here Riemannian Score-based Generative Models (RSGMs), a class of generative models extending SGMs to Riemannian manifolds. We demonstrate our approach on a variety of manifolds, and in particular with earth and climate science spherical data.

Diffusion generative models have excelled at diverse image generation and reconstruction tasks across fields. A less explored avenue is their application to discriminative tasks involving regression or classification problems. The cornerstone of modern cosmology is the ability to generate predictions for observed astrophysical fields from theory and constrain physical models from observations using these predictions. This work uses a single diffusion generative model to address these interlinked objectives -- as a surrogate model or emulator for cold dark matter density fields conditional on input cosmological parameters, and as a parameter inference model that solves the inverse problem of constraining the cosmological parameters of an input field. The model is able to emulate fields with summary statistics consistent with those of the simulated target distribution. We then leverage the approximate likelihood of the diffusion generative model to derive tight constraints on cosmology by using the Hamiltonian Monte Carlo method to sample the posterior on cosmological parameters for a given test image. Finally, we demonstrate that this parameter inference approach is more robust to small perturbations of noise to the field than baseline parameter inference networks.

We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples (x,y) from an unknown distribution on R^n times { pm 1}, whose marginal distribution on x is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss OPT+epsilon, where OPT is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors (LWE) problem. Prior hardness results are either qualitatively suboptimal or apply to restricted families of algorithms. Our techniques extend to yield near-optimal lower bounds for related problems, including ReLU regression.

We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training mean-field neural networks, maximum mean discrepancy minimization and kernel Stein discrepancy minimization. Our algorithm is based on a novel spacetime discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.

3D Gaussian Splatting (3DGS) has made significant strides in novel view synthesis but is limited by the substantial number of Gaussian primitives required, posing challenges for deployment on lightweight devices. Recent methods address this issue by compressing the storage size of densified Gaussians, yet fail to preserve rendering quality and efficiency. To overcome these limitations, we propose ProtoGS to learn Gaussian prototypes to represent Gaussian primitives, significantly reducing the total Gaussian amount without sacrificing visual quality. Our method directly uses Gaussian prototypes to enable efficient rendering and leverage the resulting reconstruction loss to guide prototype learning. To further optimize memory efficiency during training, we incorporate structure-from-motion (SfM) points as anchor points to group Gaussian primitives. Gaussian prototypes are derived within each group by clustering of K-means, and both the anchor points and the prototypes are optimized jointly. Our experiments on real-world and synthetic datasets prove that we outperform existing methods, achieving a substantial reduction in the number of Gaussians, and enabling high rendering speed while maintaining or even enhancing rendering fidelity.

3D Gaussian Splatting (3DGS) has recently revolutionized radiance field reconstruction, achieving high quality novel view synthesis and fast rendering speed without baking. However, 3DGS fails to accurately represent surfaces due to the multi-view inconsistent nature of 3D Gaussians. We present 2D Gaussian Splatting (2DGS), a novel approach to model and reconstruct geometrically accurate radiance fields from multi-view images. Our key idea is to collapse the 3D volume into a set of 2D oriented planar Gaussian disks. Unlike 3D Gaussians, 2D Gaussians provide view-consistent geometry while modeling surfaces intrinsically. To accurately recover thin surfaces and achieve stable optimization, we introduce a perspective-accurate 2D splatting process utilizing ray-splat intersection and rasterization. Additionally, we incorporate depth distortion and normal consistency terms to further enhance the quality of the reconstructions. We demonstrate that our differentiable renderer allows for noise-free and detailed geometry reconstruction while maintaining competitive appearance quality, fast training speed, and real-time rendering. Our code will be made publicly available.

Recently single-view 3D generation via Gaussian splatting has emerged and developed quickly. They learn 3D Gaussians from 2D RGB images generated from pre-trained multi-view diffusion (MVD) models, and have shown a promising avenue for 3D generation through a single image. Despite the current progress, these methods still suffer from the inconsistency jointly caused by the geometric ambiguity in the 2D images, and the lack of structure of 3D Gaussians, leading to distorted and blurry 3D object generation. In this paper, we propose to fix these issues by GS-RGBN, a new RGBN-volume Gaussian Reconstruction Model designed to generate high-fidelity 3D objects from single-view images. Our key insight is a structured 3D representation can simultaneously mitigate the afore-mentioned two issues. To this end, we propose a novel hybrid Voxel-Gaussian representation, where a 3D voxel representation contains explicit 3D geometric information, eliminating the geometric ambiguity from 2D images. It also structures Gaussians during learning so that the optimization tends to find better local optima. Our 3D voxel representation is obtained by a fusion module that aligns RGB features and surface normal features, both of which can be estimated from 2D images. Extensive experiments demonstrate the superiority of our methods over prior works in terms of high-quality reconstruction results, robust generalization, and good efficiency.

Generative models are a promising tool to produce cosmological simulations but face significant challenges in scalability, physical consistency, and adherence to domain symmetries, limiting their utility as alternatives to N-body simulations. To address these limitations, we introduce a score-based generative model with an equivariant graph neural network that simulates gravitational clustering of galaxies across cosmologies starting from an informed prior, respects periodic boundaries, and scales to full galaxy counts in simulations. A novel topology-aware noise schedule, crucial for large geometric graphs, is introduced. The proposed equivariant score-based model successfully generates full-scale cosmological point clouds of up to 600,000 halos, respects periodicity and a uniform prior, and outperforms existing diffusion models in capturing clustering statistics while offering significant computational advantages. This work advances cosmology by introducing a generative model designed to closely resemble the underlying gravitational clustering of structure formation, moving closer to physically realistic and efficient simulators for the evolution of large-scale structures in the universe.

3D Gaussian Splatting (3DGS) has emerged as a promising representation for photorealistic rendering of 3D scenes. However, its high storage requirements pose significant challenges for practical applications. We observe that Gaussians exhibit distinct roles and characteristics that are analogous to traditional artistic techniques -- Like how artists first sketch outlines before filling in broader areas with color, some Gaussians capture high-frequency features like edges and contours; While other Gaussians represent broader, smoother regions, that are analogous to broader brush strokes that add volume and depth to a painting. Based on this observation, we propose a novel hybrid representation that categorizes Gaussians into (i) Sketch Gaussians, which define scene boundaries, and (ii) Patch Gaussians, which cover smooth regions. Sketch Gaussians are efficiently encoded using parametric models, leveraging their geometric coherence, while Patch Gaussians undergo optimized pruning, retraining, and vector quantization to maintain volumetric consistency and storage efficiency. Our comprehensive evaluation across diverse indoor and outdoor scenes demonstrates that this structure-aware approach achieves up to 32.62% improvement in PSNR, 19.12% in SSIM, and 45.41% in LPIPS at equivalent model sizes, and correspondingly, for an indoor scene, our model maintains the visual quality with 2.3% of the original model size.

3D Gaussian Splatting (3DGS) has emerged as a mainstream for novel view synthesis, leveraging continuous aggregations of Gaussian functions to model scene geometry. However, 3DGS suffers from substantial memory requirements to store the multitude of Gaussians, hindering its practicality. To address this challenge, we introduce GaussianSpa, an optimization-based simplification framework for compact and high-quality 3DGS. Specifically, we formulate the simplification as an optimization problem associated with the 3DGS training. Correspondingly, we propose an efficient "optimizing-sparsifying" solution that alternately solves two independent sub-problems, gradually imposing strong sparsity onto the Gaussians in the training process. Our comprehensive evaluations on various datasets show the superiority of GaussianSpa over existing state-of-the-art approaches. Notably, GaussianSpa achieves an average PSNR improvement of 0.9 dB on the real-world Deep Blending dataset with 10times fewer Gaussians compared to the vanilla 3DGS. Our project page is available at https://noodle-lab.github.io/gaussianspa/.

Weak perturbations of equal size can produce terminal Wasserstein errors of different orders. For Gaussian-smoothed infinitely divisible laws, we differentiate distribution functions with respect to variance-weighted Lévy characteristics, including distributional directions generated by moving atoms. We obtain first- and second-order expansions uniform over Lipschitz tests. Gaussian analyticity identifies the signed second coefficient of the Wasserstein-1 distance when the first response is nonzero. When it vanishes, a finite second displacement moment yields a quadratic response, strictly positive for nonzero displacement dispersion. For local balanced remeshing, the error is comparable to the grid-alignment variance, giving sharp quadratic grid rates for compactly supported densities and nonaligned atomic sequences. We derive metric speed and length along admissible non-atomic curves and a response-based linear program with a certified oracle gap and consistent quadrature. We also establish the sharp vanishing-smoothing transition and a multivariate second-order expansion. The Supplement treats state-dependent responses and further stability estimates.

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there are finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, real-world data is often not naturally posed in this setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions. Then, we convert each GP into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications. We release the code of FunBaT at https://github.com/xuangu-fang/Functional-Bayesian-Tucker-Decomposition.

3D Gaussian Splatting (3D-GS) has recently attracted great attention with real-time and photo-realistic renderings. This technique typically takes perspective images as input and optimizes a set of 3D elliptical Gaussians by splatting them onto the image planes, resulting in 2D Gaussians. However, applying 3D-GS to panoramic inputs presents challenges in effectively modeling the projection onto the spherical surface of {360^circ} images using 2D Gaussians. In practical applications, input panoramas are often sparse, leading to unreliable initialization of 3D Gaussians and subsequent degradation of 3D-GS quality. In addition, due to the under-constrained geometry of texture-less planes (e.g., walls and floors), 3D-GS struggles to model these flat regions with elliptical Gaussians, resulting in significant floaters in novel views. To address these issues, we propose 360-GS, a novel 360^{circ} Gaussian splatting for a limited set of panoramic inputs. Instead of splatting 3D Gaussians directly onto the spherical surface, 360-GS projects them onto the tangent plane of the unit sphere and then maps them to the spherical projections. This adaptation enables the representation of the projection using Gaussians. We guide the optimization of 360-GS by exploiting layout priors within panoramas, which are simple to obtain and contain strong structural information about the indoor scene. Our experimental results demonstrate that 360-GS allows panoramic rendering and outperforms state-of-the-art methods with fewer artifacts in novel view synthesis, thus providing immersive roaming in indoor scenarios.

Studies show that the model-independent, fully non-perturbative covariant cosmographic approach is suitable for analyzing the local Universe (zlesssim 0.1). However, accurately characterizing large and inhomogeneous mass distributions requires the fourth-order term in the redshift expansion of the covariant luminosity distance d_L(z,n). We calculate the covariant snap parameter S and its spherical harmonic multipole moments using the matter expansion tensor and the evolution equations for lightray bundles. The fourth-order term adds 36 degrees of freedom, since the highest independent multipole of the snap is the 32-pole (dotriacontapole) (ell=5). Including this term helps to de-bias estimations of the covariant deceleration parameter. Given that observations suggest axially symmetric anisotropies in the Hubble diagram for z lesssim 0.1 and theory shows that only a subset of multipoles contributes to the signal, we demonstrate that only 12 degrees of freedom are needed for a model-independent description of the local universe. We use an analytical axisymmetric model of the local Universe, with data that matches the Zwicky Transient Facility survey, in order to provide a numerical example of the amplitude of the snap multipoles and to forecast precision.

We study the bi-parameter local linearization of the one-dimensional nonlinear stochastic wave equation driven by a Gaussian noise, which is white in time and has a spatially homogeneous covariance structure of Riesz-kernel type. We establish that the second-order increments of the solution can be approximated by those of the corresponding linearized wave equation, modulated by the diffusion coefficient. These findings extend the previous results of Huang et al. HOO2024, which addressed the case of space-time white noise. As applications, we analyze the quadratic variation of the solution and construct a consistent estimator for the diffusion parameter.

Feed-forward 3D Gaussian Splatting methods reconstruct a scene from posed or pose-free images in a single forward pass, yet current approaches predict one Gaussian per input pixel, tying the representation budget to camera resolution rather than scene complexity. A flat wall and a richly textured object thus produce equally many Gaussians despite very different geometric needs. We propose ZipSplat, a token-based feed-forward model that decouples Gaussian placement from the pixel grid. A multi-view backbone extracts dense visual tokens, and k-means clustering compresses them into a compact set of scene tokens. Cross- and self-attention refine these tokens, and a lightweight MLP decodes each into a group of Gaussians with unconstrained 3D positions. Because clustering is applied at inference, a single trained model spans the quality-efficiency curve without retraining. ZipSplat operates without ground-truth poses or intrinsics, yet sets a new state of the art on DL3DV and RealEstate10K with {sim}6{times} fewer Gaussians than pixel-aligned methods, surpassing the best pose-free baseline by 2.1dB and 1.2dB PSNR, respectively. It further generalizes zero-shot to Mip-NeRF360 and ScanNet++, outperforming all comparable baselines. Our project page is at {https://veichta.com/zipsplat{https://veichta.com/zipsplat}}.

We give the first efficient algorithm for learning halfspaces in the testable learning model recently defined by Rubinfeld and Vasilyan (2023). In this model, a learner certifies that the accuracy of its output hypothesis is near optimal whenever the training set passes an associated test, and training sets drawn from some target distribution -- e.g., the Gaussian -- must pass the test. This model is more challenging than distribution-specific agnostic or Massart noise models where the learner is allowed to fail arbitrarily if the distributional assumption does not hold. We consider the setting where the target distribution is Gaussian (or more generally any strongly log-concave distribution) in d dimensions and the noise model is either Massart or adversarial (agnostic). For Massart noise, our tester-learner runs in polynomial time and outputs a hypothesis with (information-theoretically optimal) error opt + epsilon for any strongly log-concave target distribution. For adversarial noise, our tester-learner obtains error O(opt) + epsilon in polynomial time when the target distribution is Gaussian; for strongly log-concave distributions, we obtain O(opt) + epsilon in quasipolynomial time. Prior work on testable learning ignores the labels in the training set and checks that the empirical moments of the covariates are close to the moments of the base distribution. Here we develop new tests of independent interest that make critical use of the labels and combine them with the moment-matching approach of Gollakota et al. (2023). This enables us to simulate a variant of the algorithm of Diakonikolas et al. (2020) for learning noisy halfspaces using nonconvex SGD but in the testable learning setting.

Conformal symmetries, i.e.\ coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is equivariant under general conformal transformations. To achieve this, we lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit a known correspondence between conformal transformations of flat space and isometric transformations on the AdS space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geometric deep learning literature. We employ message-passing layers conditioned on the proper distance, yielding a computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.

Data-driven learning approaches for physics simulation, sometimes referred to as world models, have emerged as promising alternatives to traditional physics simulators due to their differentiable nature. Prior work has demonstrated impressive results in predicting the motions of rigid and non-rigid objects in complex scenes involving multiple interacting bodies. However, these models are typically trained in simulated environments because obtaining perfect state information such as complete scene point clouds and point correspondences over time is challenging in real-world settings. This reliance on synthetic data can limit their applicability when the sim-to-real gap is large. In this work, we aim to overcome these limitations by introducing a novel framework for training neural object dynamics models directly from unlabeled real-world videos. Specifically, we propose to learn a particle-based dynamics model compatible with a Gaussian splatting framework, which operates on dense particles derived from Gaussians (i.e., particles with scales and rotations) and predicts their position and rotation changes over time. The model is trained via rendering supervision, enabling learning from real-world videos without requiring particle-level labeled states. Our model operates directly on dense Gaussians without relying on heuristic subsampling anchor points. To enable this study, we also present a real-world dataset consisting of about 500 videos capturing diverse object interactions.

A major goal of ongoing and future cosmological surveys of the large-scale structure is to measure local type primordial non-Gaussianity in the galaxy power spectrum through the scale-dependent bias. General relativistic effects have been shown to be degenerate with this measurement, therefore requiring a non-Newtonian approach. In this work, we develop a consistent framework to compute integrated effects, including lensing convergence, time delay, and integrated Sachs--Wolfe, along with the local relativistic projection and wide-separation corrections in the multipoles of the power spectrum. We show that, for a Euclid-like Hα-line galaxy survey and a MegaMapper-like Lyman-break galaxy survey, ignoring these effects leads to a bias on the best fit measurement of the amplitude of primordial non-Gaussianity, f_{rm NL}, of around 3,σ and 20 , σ respectively. When we include these corrections, the uncertainty in our knowledge of the luminosity function leads to further uncertainty in our measurement of f_{rm NL}. In this work, we show that this degeneracy can be partly mitigated by using a bright-faint multi-tracer analysis, where the observed galaxy sample is subdivided into two separate populations based on luminosity, which provides a 15--20% improvement on the forecasted constraints of local type f_{rm NL}. In addition, we present a novel calculation of the full multi-tracer covariance with the inclusion of wide-separation corrections~-- all of these results are implemented in the Python code CosmoWAP.

Automatic car damage detection has been a topic of significant interest for the auto insurance industry as it promises faster, accurate, and cost-effective damage assessments. However, few works have gone beyond 2D image analysis to leverage 3D reconstruction methods, which have the potential to provide a more comprehensive and geometrically accurate representation of the damage. Moreover, recent methods employing 3D representations for novel view synthesis, particularly 3D Gaussian Splatting (3D-GS), have demonstrated the ability to generate accurate and coherent 3D reconstructions from a limited number of views. In this work we introduce an automatic car damage detection pipeline that performs 3D damage segmentation by up-lifting 2D masks. Additionally, we propose a simple yet effective learning-free approach for single-view 3D-GS segmentation. Specifically, Gaussians are projected onto the image plane using camera parameters obtained via Structure from Motion (SfM). They are then filtered through an algorithm that utilizes Z-buffering along with a normal distribution model of depth and opacities. Through experiments we found that this method is particularly effective for challenging scenarios like car damage detection, where target objects (e.g., scratches, small dents) may only be clearly visible in a single view, making multi-view consistency approaches impractical or impossible. The code is publicly available at: https://github.com/DragosChileban/CrashSplat.

Gaussian process state-space models (GPSSMs) provide a principled and flexible approach to modeling the dynamics of a latent state, which is observed at discrete-time points via a likelihood model. However, inference in GPSSMs is computationally and statistically challenging due to the large number of latent variables in the model and the strong temporal dependencies between them. In this paper, we propose a new method for inference in Bayesian GPSSMs, which overcomes the drawbacks of previous approaches, namely over-simplified assumptions, and high computational requirements. Our method is based on free-form variational inference via stochastic gradient Hamiltonian Monte Carlo within the inducing-variable formalism. Furthermore, by exploiting our proposed variational distribution, we provide a collapsed extension of our method where the inducing variables are marginalized analytically. We also showcase results when combining our framework with particle MCMC methods. We show that, on six real-world datasets, our approach can learn transition dynamics and latent states more accurately than competing methods.

Videos of robots interacting with objects encode rich information about the objects' dynamics. However, existing video prediction approaches typically do not explicitly account for the 3D information from videos, such as robot actions and objects' 3D states, limiting their use in real-world robotic applications. In this work, we introduce a framework to learn object dynamics directly from multi-view RGB videos by explicitly considering the robot's action trajectories and their effects on scene dynamics. We utilize the 3D Gaussian representation of 3D Gaussian Splatting (3DGS) to train a particle-based dynamics model using Graph Neural Networks. This model operates on sparse control particles downsampled from the densely tracked 3D Gaussian reconstructions. By learning the neural dynamics model on offline robot interaction data, our method can predict object motions under varying initial configurations and unseen robot actions. The 3D transformations of Gaussians can be interpolated from the motions of control particles, enabling the rendering of predicted future object states and achieving action-conditioned video prediction. The dynamics model can also be applied to model-based planning frameworks for object manipulation tasks. We conduct experiments on various kinds of deformable materials, including ropes, clothes, and stuffed animals, demonstrating our framework's ability to model complex shapes and dynamics. Our project page is available at https://gs-dynamics.github.io.

In physics and engineering literature, the distribution of the excursion-above-zero time distribution (exceedance distribution) for a stationary Gaussian process has been approximated by a stationary switching process with independently distributed switching times. The approach matched the covariance of the clipped Gaussian process with the one for the stationary switching process and the distribution of the latter was used as the so-called independent interval approximation (IIA). The approach successfully assessed the persistency exponent for many physically important processes but left an unanswered question when such an approach leads to a mathematically meaningful and proper exceedance distribution. Here we address this question by proposing an alternative matching of the expected values of the clipped Slepian process and the corresponding switched process initiated at the origin. The method has allowed resolving the mathematical correctness of the matching method for a large subclass of the Gaussian processes with monotonic covariance, for which we provide a sufficient condition for the validity of the IIA. Within this class, the IIA produces a valid distribution for the excursion time and is represented in an explicit stochastic form that connects directly to the covariance of the underlying Gaussian process. We compare the excursion level distributions as well as the corresponding persistency exponents obtained through the IIA method with numerically computed exact distributions, and the simulated distribution for several important Gaussian models. We also argue that for stationary Gaussian processes with a non-monotonic covariance, the IIA fails and should not be used.

There is a growing gap between the impressive results of deep image generative models and classical algorithms that offer theoretical guarantees. The former suffer from mode collapse or memorization issues, limiting their application to scientific data. The latter require restrictive assumptions such as log-concavity to escape the curse of dimensionality. We partially bridge this gap by introducing conditionally strongly log-concave (CSLC) models, which factorize the data distribution into a product of conditional probability distributions that are strongly log-concave. This factorization is obtained with orthogonal projectors adapted to the data distribution. It leads to efficient parameter estimation and sampling algorithms, with theoretical guarantees, although the data distribution is not globally log-concave. We show that several challenging multiscale processes are conditionally log-concave using wavelet packet orthogonal projectors. Numerical results are shown for physical fields such as the varphi^4 model and weak lensing convergence maps with higher resolution than in previous works.

The increasing demand for virtual reality applications has highlighted the significance of crafting immersive 3D assets. We present a text-to-3D 360^{circ} scene generation pipeline that facilitates the creation of comprehensive 360^{circ} scenes for in-the-wild environments in a matter of minutes. Our approach utilizes the generative power of a 2D diffusion model and prompt self-refinement to create a high-quality and globally coherent panoramic image. This image acts as a preliminary "flat" (2D) scene representation. Subsequently, it is lifted into 3D Gaussians, employing splatting techniques to enable real-time exploration. To produce consistent 3D geometry, our pipeline constructs a spatially coherent structure by aligning the 2D monocular depth into a globally optimized point cloud. This point cloud serves as the initial state for the centroids of 3D Gaussians. In order to address invisible issues inherent in single-view inputs, we impose semantic and geometric constraints on both synthesized and input camera views as regularizations. These guide the optimization of Gaussians, aiding in the reconstruction of unseen regions. In summary, our method offers a globally consistent 3D scene within a 360^{circ} perspective, providing an enhanced immersive experience over existing techniques. Project website at: http://dreamscene360.github.io/

Neural implicit representations, including Neural Distance Fields and Neural Radiance Fields, have demonstrated significant capabilities for reconstructing surfaces with complicated geometry and topology, and generating novel views of a scene. Nevertheless, it is challenging for users to directly deform or manipulate these implicit representations with large deformations in the real-time fashion. Gaussian Splatting(GS) has recently become a promising method with explicit geometry for representing static scenes and facilitating high-quality and real-time synthesis of novel views. However,it cannot be easily deformed due to the use of discrete Gaussians and lack of explicit topology. To address this, we develop a novel GS-based method that enables interactive deformation. Our key idea is to design an innovative mesh-based GS representation, which is integrated into Gaussian learning and manipulation. 3D Gaussians are defined over an explicit mesh, and they are bound with each other: the rendering of 3D Gaussians guides the mesh face split for adaptive refinement, and the mesh face split directs the splitting of 3D Gaussians. Moreover, the explicit mesh constraints help regularize the Gaussian distribution, suppressing poor-quality Gaussians(e.g. misaligned Gaussians,long-narrow shaped Gaussians), thus enhancing visual quality and avoiding artifacts during deformation. Based on this representation, we further introduce a large-scale Gaussian deformation technique to enable deformable GS, which alters the parameters of 3D Gaussians according to the manipulation of the associated mesh. Our method benefits from existing mesh deformation datasets for more realistic data-driven Gaussian deformation. Extensive experiments show that our approach achieves high-quality reconstruction and effective deformation, while maintaining the promising rendering results at a high frame rate(65 FPS on average).

We study the problem of learning generative models for discrete sequences in a continuous embedding space. Whereas prior approaches typically operate in Euclidean space or on the probability simplex, we instead work on the sphere mathbb S^{d-1}. There the von Mises-Fisher (vMF) distribution induces a natural noise process and admits a closed-form conditional score. The conditional velocity is in general intractable. Exploiting the radial symmetry of the vMF density we reduce the continuity equation on mathbb S^{d-1} to a scalar ODE in the cosine similarity, whose unique bounded solution determines the velocity. The marginal velocity and marginal score on (mathbb S^{d-1})^L both decompose into posterior-weighted tangent sums that differ only by per-token scalar weights. This gives access to both ODE and predictor-corrector (PC) sampling. The posterior is the only learned object, trained by a cross-entropy loss. Experiments compare the vMF path against geodesic and Euclidean alternatives. The combination of vMF and PC sampling significantly improves results on Sudoku and language modeling.

Gaussian processes (GP) are one of the most successful frameworks to model uncertainty. However, GP optimization (e.g., GP-UCB) suffers from major scalability issues. Experimental time grows linearly with the number of evaluations, unless candidates are selected in batches (e.g., using GP-BUCB) and evaluated in parallel. Furthermore, computational cost is often prohibitive since algorithms such as GP-BUCB require a time at least quadratic in the number of dimensions and iterations to select each batch. In this paper, we introduce BBKB (Batch Budgeted Kernel Bandits), the first no-regret GP optimization algorithm that provably runs in near-linear time and selects candidates in batches. This is obtained with a new guarantee for the tracking of the posterior variances that allows BBKB to choose increasingly larger batches, improving over GP-BUCB. Moreover, we show that the same bound can be used to adaptively delay costly updates to the sparse GP approximation used by BBKB, achieving a near-constant per-step amortized cost. These findings are then confirmed in several experiments, where BBKB is much faster than state-of-the-art methods.

We identify a key limitation of existing feed-forward 3DGS methods: Gaussian attributes are typically predicted in a radial-distance-normalized space that canonicalizes a global radial statistic of the reconstructed geometry, while the spacing between neighboring Gaussian centers can still vary substantially across scene scales and camera configurations. This mismatch induces systematic shifts in Gaussian-scale targets, leading to ambiguous supervision. To address this, we propose SplatAnything with Gaussian Spacing Normalization (GSNorm), which canonicalizes a robust scene-level statistic of Gaussian spacing to provide a more consistent spatial reference for Gaussian prediction. To rigorously assess our method, we further establish a systematic benchmark spanning diverse datasets, input sparsity levels, and challenging extrapolated viewpoints. Extensive experiments demonstrate that SplatAnything achieves state-of-the-art performance, with particularly strong gains under substantial variations in scene scale and viewpoint.