3D Gaussian Splatting

We propose PACE-GGM, a data-adaptive differentially private method for covariance estimation that concentrates its privacy budget on the most informative entries of the empirical covariance matrix, rather than perturbing all entries. This applies in the natural setting where the modeler supplies separate bounds for each variable, so that individual entries can be measured with less noise than the full matrix. In each round, our method selects a poorly approximated entry, measures it using the Gaussian mechanism, and then reconstructs a full covariance matrix using a maximum-entropy reconstruction objective, leading to a Gaussian graphical model structure. Experiments on diverse real-world datasets demonstrate consistent improvements in estimation error with respect to the Gaussian mechanism and other baselines, particularly in high-dimensional and low-to-moderate privacy regimes.

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.

Reliable uncertainty estimation is critical for deploying monocular depth deep neural networks (DNNs) in safety-critical robotic systems. Conventional uncertainty methods such as ensembles and sampling-based approaches require multiple inferences per image, incurring substantial compute and memory overhead. Moreover, uncertainty predicted from a single image misses out on measuring disagreement between predictions across views of the same region. We propose Uncertainty from Motion* (UfM*), an uncertainty estimation algorithm that measures multiview disagreement efficiently by comparing previous and current views using a compact Gaussian mixture, requiring only a single DNN inference per image. Using Gaussians to compute multiview disagreement is not only more compute- and memory-efficient than a prior approach using a point cloud, but also improves uncertainty by measuring disagreement across regions of 3D space. UfM* paired with aleatoric uncertainty improves expected calibration error by 24-28% compared to an ensemble, while requiring only 3% of the energy and 0.02% of the memory on 100 out-of-distribution ScanNet sequences. We demonstrate UfM* consumes only 63 mJ per 224x224 image while running real-time at 30 FPS on an Arm Cortex-A76 CPU onboard a miniature energy-constrained robot, highlighting that measuring multiview disagreement using Gaussians enables efficient uncertainty for resource-constrained robotic systems.

Of $1{,}536$ Gaussian release covariances we tested for single-layer hidden-state privacy, zero achieve both moderate utility and moderate privacy against an adaptive retrieval attacker. We prove a complementary Fisher-ball lower bound: every full-rank Gaussian release at $O(1)$ Fisher utility admits a direction whose Mahalanobis signal grows linearly in hidden width, ruling out uniform Gaussian safety in the class and matching the empirical empty middle. The diagonal inverse-Fisher release $Σ^\star_{\mathrm{diag}}(\mathcal{K}) = (2\mathcal{K}/d)\,\mathrm{diag}(1/F_{ii})$ is the unique minimax-optimal diagonal mechanism at first-order KL budget $\mathcal{K}$ and the only release with worst-attacker top-1 $\le 0.001$ at every point of a 32 model-layer grid, but it sits on a privacy/utility edge rather than filling the middle. A generalized-eigen mechanism reaching $13\times$ Pareto reduction under Euclidean retrieval collapses to $100\%$ top-1 under the adaptive Mahalanobis attacker, and a full-trajectory sequence inverter recovers $94\%$ of clean GPT-2 prefixes but $0\%$ under $Σ_{\mathrm{diag}}$. A split-memory transformer trained from scratch reaches $G_{\mathrm{Mah}} \in [20, 33]$ at 90M and maintains a $6$--$24\times$ advantage over same-budget GPT baselines from 30M to 1B at a fixed-token language-modeling loss penalty; pretrained models top out at 9.3. These results reframe hidden-state release from mechanism-design within the Gaussian class to architecture or release co-design.

Recent feed-forward 3D gaussian splatting methods have made dramatic progress on individual aspects of 3D scene reconstruction, but no existing method jointly addresses dynamic content, multi-view input, and unknown camera poses in a single feed-forward pass. Methods that handle dynamics either require accurate camera poses or accept only monocular input; pose-free multi-view methods address only static scenes; and per-scene optimization methods bridge some of these gaps but at minutes-to-hours cost per scene. We introduce NoPo4D, the first feed-forward system that addresses this empty quadrant. Building on a pretrained geometry backbone and recent 4D Gaussian frameworks, NoPo4D introduces a velocity decomposition that splits Gaussian motion into per-pixel image-plane shifts and depth changes, allowing direct supervision from pseudo ground-truth optical flow on the 2D component. This sidesteps both the differentiable rendering that couples prior posed methods to pose accuracy and the 3D motion ground truth that prior pose-free methods require. The system is rounded out by a bidirectional motion encoder for cross-view and cross-frame feature aggregation, and view-dependent opacity that mitigates cross-view and cross-timestep Gaussian misalignments. On four multi-view dynamic benchmarks, NoPo4D consistently outperforms prior feed-forward baselines, and with an optional post-optimization stage surpasses per-scene optimization methods, while running orders of magnitude faster.

Graph Neural Networks (GNNs) have become the de facto standard for learning on relational data. While traditional GNNs' message passing is well suited for vector-valued node features, there are cases in which node features are better represented by probability distributions than real vectors. Concretely, when node features are Gaussians, characterized by a mean and a covariance matrix, naively concatenating their parameters into a single vector and applying standard message passing discards the geometric and algebraic structure that governs means and covariances. We propose Gaussian Sheaf Neural Networks (GSNNs), a principled framework that incorporates these inductive biases into graph-based learning. Building on the theory of cellular sheaves, we derive a new Laplacian operator that generalizes the sheaf Laplacian to this setting and preserves its key properties. We complement our theoretical contributions with experiments on synthetic and real-world data that illustrate the practical relevance of GSNNs.

We study the task of agnostic learning of multiclass linear classifiers under the Gaussian distribution. Given labeled examples $(x, y)$ from a distribution over $\mathbb{R}^d \times [k]$, with Gaussian $x$-marginal, the goal is to output a hypothesis whose error is comparable to that of the best $k$-class linear classifier. While the binary case $k=2$ has a well-developed algorithmic theory, much less is known for $k \ge 3$. Even for $k=3$, prior robust algorithms incur exponential dependence on the inverse of the desired accuracy in both complexity and representation size. In this work, we develop new structural results for multiclass linear classifiers and use them to design fully polynomial-time robust learners with dimension-independent error guarantees. Our first result shows that the standard multiclass perceptron algorithm requires super-polynomially many samples and updates, even with clean labels and Gaussian marginals, revealing a basic obstruction absent in the binary case. Our main positive result is a pairwise improper-learning framework which yields an efficient learner with error $\widetilde O(k^{3/2}\sqrt{\mathrm{opt}})+ε$ for general $k$. Additionally, we develop a sharper localization-based framework which leads to error $O(\mathrm{opt})+ε$ for $k=3$, and error $\mathrm{poly}(k)\mathrm{opt}+ε$ for geometrically regular $k$-class linear classifiers.

Gaussian processes (GPs) offer a principled probabilistic model over functions, but exact inference is restricted to the linear-Gaussian regime. We establish an explicit equivalence between GPs and a class of linear diffusion models, recasting predictive sampling as an ODE with closed-form Gaussian dynamics and a likelihood-dependent guidance term that admits a simple Monte Carlo approximation. In the linear-Gaussian setting, we recover standard GP conditioning exactly; beyond conjugacy, the same machinery handles any conditioning statement admitting point-wise likelihood evaluation -- including non-linear physics, and, for the first time, natural language via large language models. Whitening isolates the irreducible non-Gaussian dynamics, minimising Wasserstein-2 transport cost and eliminating numerical stiffness. The result is a general-purpose GP inference scheme requiring no bespoke derivations. Together, these results provide a general mechanism for incorporating the full richness of real-world knowledge as conditioning information, opening a new frontier for the probabilistic modelling of real-world problems.

Existing 3D clothed avatar reconstruction methods achieve high visual fidelity but ignore geometric structure and physical plausibility. They either model clothed humans as a single deformable surface or attempt garment disentanglement without enforcing geometric constraints, resulting in ambiguous garment boundaries and no control over stacking or layer ordering. To address these limitations, we introduce DAMA (Disentangled body-Anchored Gaussians for Controllable Multi-layered Avatars), a 3D avatar reconstruction method that produces physically plausible clothed avatars through a dedicated representation and reconstruction method. At the representation level, we bind Gaussians to SMPL-X faces using barycentric in-plane coordinates and a positive normal offset. Based on this parameterization, the reconstruction method lifts 2D segmentations to body-anchored Gaussians, refines layers using topology-guided correction, and jointly optimizes geometry and appearance. DAMA is the first Gaussian avatar reconstruction method from multi-view images to achieve physically plausible layering, clean garment separation, and explicit stacking control. On the full 4D-DRESS dataset (82 scans), it achieves state-of-the-art performance in geometry reconstruction, garment separation, penetration rate, and penetration depth. The representation further supports user-defined garment reordering and fast conversion of body-conforming garments to simulation-ready meshes. Project Page: https://danieleskandar.github.io/dama/

2D Gaussian splatting provides an efficient explicit representation for image reconstruction, but existing methods still require costly per-image iterative optimization or rely on handcrafted priors for primitive allocation. We present AIR, a self-supervised feed-forward framework that amortizes iterative Gaussian fitting into a single network pass, eliminating per-image test-time optimization. AIR adopts a stage-wise residual architecture that progressively predicts additional Gaussian primitives from reconstruction residuals, together with an explicit Stage Control mechanism that activates new primitives only in under-reconstructed regions. A Predict--Optimize--Distill training strategy stabilizes multi-stage prediction by distilling short-horizon optimized Gaussian increments back into the predictor. The stabilized predictor is then jointly finetuned across stages and equipped with an image-adaptive quantizer for compact Gaussian storage. Experiments on Kodak and DIV2K show that AIR achieves better reconstruction quality than representative Gaussian-based baselines while reducing encoding time to 160--300,ms. Code: https://github.com/whoiszzj/AIR.git

Vision-language-action (VLA) policies have advanced language-conditioned robotic manipulation by transferring semantic priors from pretrained vision-language models to action generation. However, standard action-imitation learning often lacks sufficient modeling of explicit 3D spatial information, dense geometric supervision, and future environment evolution, all critical for precise robotic interaction. To address this, we propose \textbf{GaussianDream}, a feed-forward 3D Gaussian world-model plug-in. Specifically, we introduce learnable GaussianDream Queries in the encoder, enabling the model to capture current-frame 3D spatial structure and short-horizon future evolution. During training, the latent GaussianDream prefix is processed by a static reconstruction head and a future prediction head to produce current 3D Gaussian scene states and future Gaussian evolution states. The current branch is supervised by RGB rendering and depth, while the future branch uses future RGB, depth, and pseudo 3D scene-flow signals. During inference, GaussianDream discards all auxiliary heads and retains only the learned prefix to condition action generation, without test-time Gaussian reconstruction or future prediction. Experimental results demonstrate that GaussianDream achieves state-of-the-art performance across multiple robotic manipulation benchmarks, reaching \textbf{98.4%} on LIBERO, \textbf{54.8%} on RoboCasa Human-50, and \textbf{50.0%} on real-robot tasks. Compared with existing 3D-enhanced VLA methods, GaussianDream achieves strong accuracy while providing higher inference efficiency than video-based world-model approaches.

Latent Gaussian models (LGMs) are a popular class of Bayesian hierarchical models that include Gaussian processes, as well as certain spatial models and mixed-effect models. Efficient Bayesian inference of LGMs often requires marginalizing out the latent variables. For LGMs with a non-Gaussian likelihood, exact marginalization is not possible and a popular approach is to do approximate marginalization with an integrated Laplace approximation (ILA). Using ILA produces an approximate posterior which, in some settings, can differ significantly from the correct posterior, which impacts downstream applications. We propose an importance sampling scheme to correct the error introduced by ILA. By increasing the number of samples in importance sampling, the posterior with ILA converges to the correct posterior. This idea is realized with various techniques, including pseudo-marginalization, quasi-Monte Carlo and randomized quasi-Monte Carlo. We implement our methods in an automatic differentiation framework to support gradient-based algorithms when doing inference on the hyperparameters. For the latter, we specifically consider the use of Hamiltonian Monte Carlo. We demonstrate the benefits of reduced error in various applied models.

JEPAs often regularize one-view embeddings toward an isotropic Gaussian, implicitly baking Euclidean symmetry into the representation. We show that this is not merely a benign default. For a known structured downstream geometry $H\succ0$, the minimax and maximum-entropy covariance under a Hamiltonian energy budget is $(c/d)H^{-1}$, and Euclidean isotropy incurs a closed-form price of isotropy. More importantly, when the downstream geometry is unknown, no geometry-independent fixed marginal target is canonical: every fixed covariance shape can be maximally misaligned for some structured geometry. We further show that even oracle one-view marginals do not identify the JEPA view-to-view predictive coupling. These results suggest that the structural bias in JEPAs should enter the cross-view coupling rather than a fixed encoder marginal. We instantiate this principle with \textbf{HamJEPA}, which encodes each view as a phase-space state $(q,p)$ and predicts view-to-view transitions with a learned Hamiltonian leapfrog map, while non-isotropic scale and spectral floors prevent collapse. In a deliberately headless token protocol, HamJEPA improves over SIGReg on CIFAR-100 by $+4.89$ kNN@20 and $+3.52$ linear-probe points at 30 epochs, and by $+6.45$ kNN@20 and $+10.64$ linear-probe points at 80 epochs, while a matched MLP predictor ablation shows that the symplectic coupling is the ingredient driving the neighborhood-geometry gain. On ImageNet-100, HamJEPA-$q$ improves by $+4.82$ kNN@20 and $+7.52$ linear-probe points at 45 epochs.

Reconstructing large-scale urban scenes from sparse aerial views is a crucial yet challenging task. Due to biased top-down and shallow-oblique camera poses, sparse aerial captures exhibit strong evidence imbalance: roofs and open regions are repeatedly observed, while facades, distant buildings, and occluded structures receive little multi-view support. Existing feed-forward 3D Gaussian Splatting methods directly regress a deterministic representation from sparse inputs, but this often leads to ghosting, melted facades, and stretched textures. Recent pseudo-view and video-based generative reconstruction methods use additional supervision or generative priors. However, they often lack a clear separation between observed geometry and prior-driven content, which can lead to plausible but inconsistent structures. We propose AnyCity, an observation-grounded generative reconstruction framework for sparse aerial urban scenes. AnyCity first predicts an observation-supported geometry latent to anchor reliable structures, and then uses scaffold-conditioned aerial completion tokens to predict a gated residual update for weakly constrained content before Gaussian decoding. During training, dense-to-sparse distillation transfers structural cues from dense-view reconstruction, while an aerial-adapted video diffusion prior provides fine-grained urban appearance cues through gated token conditioning. Observation-preserving objectives keep the refined representation consistent with input-supported geometry. At inference time, AnyCity reconstructs the final 3D Gaussian scene from sparse aerial views in a single feed-forward pass, achieving coherent urban novel-view synthesis with second-level inference. Experiments on synthetic, aerial-domain, UAV-textured, and real-world scenes show consistent improvements over feed-forward baselines.

3D Lookup Tables (3D LUTs) are widely used for color mapping, but their grid-based representation requires discretizing the RGB space, leading to a capacity-memory trade-off that becomes prohibitive when storing large numbers of LUTs. Recent approaches adopt implicit neural representations to improve scalability, yet their black-box nature limits interpretability and hinders intuitive, localized editing. In this paper, we propose Gaussian LUT (GLUT), a continuous and explicit color representation that models color transformations using a set of learnable 3D Gaussian primitives. By avoiding fixed-resolution grids, GLUT achieves flexible representational capacity while maintaining a compact memory footprint. Its explicit, spatially localized formulation further enables both accurate modeling and interpretability. Building on this representation, we introduce a compact conditional generator (CGLUT) that predicts GLUT parameters for multiple LUT instances, encoding diverse color styles in a single framework to enable smooth and controllable LUT style blending. Moreover, GLUT supports efficient, user-friendly editing by allowing localized adjustments to specific color regions without global retraining. Experimental results demonstrate that our approach outperforms prior neural LUT representations in both accuracy and efficiency, while offering improved interpretability and interactive control.

Predictive coding (PC) offers a local and biologically grounded alternative to backpropagation in the training of artificial neural networks, yet to date, it remains slower, and performance degrades sharply as network depth increases. We trace both problems to a single simplification: current PC networks fix the precision matrix to the identity, discarding precision-weighted prediction errors that the variational derivation requires to be fast, local, and Bayesian. We close this gap by expressing predictive coding networks as deep hierarchical Gaussian filters (HGFs) and restore precision-weighted message passing, yielding dynamic uncertainty estimates and Hebbian-compatible update rules at every layer. The resulting networks can simultaneously learn activations, weights, and precisions under a single free-energy objective, with no global error signal, and resolve inference without requiring iterations or automatic differentiation. On FashionMNIST, our solution approaches backpropagation in epoch-level wall-clock cost while converging in fewer epochs, and outperforms it on online, data efficiency, and concept-drift tasks. We thus establish that closed-form variational inference with online precision learning provides a tractable foundation for deep predictive coding networks, retaining biological and interpretative advantages, without requiring iterative relaxation or global error signals.

In this paper, we establish Berry-Esseen-type bounds for federated linear stochastic approximation (LSA). Our results provide the first federated Gaussian approximations for LSA that explicitly capture communication-computation trade-offs and heterogeneity-aware error terms, quantifying the effects of local step size, number of local updates, and heterogeneity on convergence rates. We present results for both (i) constant step size regime and (ii) decreasing step size with an increasing number of local iterations, recovering the recent rates of Bonnerjee et al. [2025] as a special case. As a primary application of our results, we develop an online multiplier bootstrap procedure for inference on the last iterate, which avoids explicit estimation of the asymptotic covariance matrix, and obtain non-asymptotic validity guarantees for this procedure.

Physics-informed neural networks (PINNs) offer a mesh-free framework for solving partial differential equations (PDEs), yet training often suffers from gradient pathologies, spectral bias, and poor convergence, especially for problems with strong nonlinearity, sharp gradients, or multiscale features. We propose the Curriculum-Guided Gaussian Mixture Physics-Informed Neural Network (CGMPINN), which integrates Gaussian mixture modeling with dynamic curriculum learning. Specifically, a GMM is periodically fitted to the PDE residual distribution to quantify spatially varying learning difficulty. A smooth curriculum schedule progressively shifts training focus from easy to harder regions, while precision-based variance modulation suppresses unreliable clusters during early optimization. This dual curriculum is governed by a shared curriculum parameter and can be combined with self-adaptive loss balancing. We further establish theoretical guarantees, including sublinear convergence of the gradient norm for the induced time-varying loss, uniform equivalence between the curriculum-weighted and standard PDE losses, and a generalization bound with an explicit weighting-induced bias characterization. Experiments on six benchmark PDEs spanning elliptic, parabolic, hyperbolic, advection-dominated, and nonlinear reaction-diffusion types show that CGMPINN consistently achieves the lowest relative $L_2$ and maximum absolute errors among all compared methods, reducing relative $L_2$ error by up to 97.8% over the standard PINN at comparable cost. Our code is publicly available at https://github.com/Mathematics-Yang/CGMPINN.

Heavy-tailed distributions are prevalent in performance evaluation, network traffic, and risk modeling. This behavior poses a fundamental challenge for modern deep generative models. Standard Variational Autoencoders (VAEs) employ Gaussian decoder likelihoods and Lipschitz-constrained neural networks, a combination that is structurally incapable of producing heavy-tailed outputs: the Gaussian tail decays exponentially, and Lipschitz continuity prevents the decoder from amplifying rare events from the latent space input to sufficiently overcome this decay. We provide both a theoretical characterization of this limitation and a controlled empirical demonstration using synthetic Pareto data across a grid of tail indices $α$ $\in$ {2, 3, 5, 30} and dimensions d $\in$ {1, 5, 10}. As a solution, we replace the Gaussian decoder with a Phase-Type (PH) distribution based on Markov chains, while keeping the encoder, latent space, and training procedure identical. PH distributions allow for arbitrarily precise approximations of any positive-valued distributions, including heavy-tailed families. Experiments showed that the PH-based model reduces tail Kolmogorov-Smirnov distance by up to x6 and extreme quantile error by up to x10 compared to the Gaussian baseline for heavy-tailed data. These results demonstrate that integrating Markov chain-based distributions into the decoder of a generative model institutes a principled and practically effective solution to the heavy-tail generation problem.

3D Gaussian Splatting (3DGS) enables real-time novel view synthesis with high visual quality. However, existing methods struggle with semi-transparent specular surfaces that exhibit both complex reflections and clear transmission, often producing blurry reflections or overly occluded transmission. To address this, we present RT-Splatting, a framework that disentangles each Gaussian's geometric occupancy from its optical opacity. This factorization yields a unified surface-volume scene representation with a single set of Gaussian primitives. Our hybrid renderer interprets this representation both as a surface to capture high-frequency reflections and as a volume to preserve clear transmission. To mitigate the ambiguity in jointly optimizing reflection and transmission, we introduce Specular-Aware Gradient Gating, which suppresses misleading gradients from highly specular regions into the transmission branch, effectively reducing distracting floaters. Experiments on challenging semi-transparent scenes show that RT-Splatting achieves state-of-the-art performance, delivering high-fidelity reflections and clear transmission with real-time rendering. Moreover, our factorization naturally enables flexible scene editing. The project page is available at https://sjj118.github.io/RT-Splatting.

Future 3D semantic occupancy forecasting and motion planning are central to autonomous driving, as they require models to reason about how surrounding scenes evolve and how the ego vehicle should act. Existing occupancy world models commonly discretize scenes into latent embeddings, volumetric features, or quantized tokens, and forecast future states through fixed-step autoregressive generation. This limits temporal flexibility, obscures scene evolution, accumulates errors over long horizons, and poorly matches the continuous-time dynamics of real driving scenes. We propose GEM, a Gaussian Evolution Model for non-autoregressive occupancy world modeling, where driving scenes are represented as explicit continuous 4D Gaussian primitives with learned dynamics. Instead of rolling out future occupancy states step by step, GEM directly queries the Gaussian world representation at arbitrary timestamps and splats the corresponding conditional 3D Gaussians into semantic occupancy volumes. This enables efficient forecasting over the full horizon while retaining a compact and interpretable scene representation. By decoupling spatial geometry, temporal support, and primitive motion, GEM makes the predicted world easier to inspect, as each primitive's evolution can be followed continuously over time. The same representation also supports motion planning by predicting future ego trajectories from the learned Gaussian world. Extensive experiments show that GEM achieves state-of-the-art future semantic occupancy forecasting and strong motion planning performance, while providing flexible temporal querying.

In this paper, we derive rates of convergence in the high-dimensional central limit theorem for Polyak--Ruppert averaged iterates generated by entropy-regularized asynchronous Q-learning with linear function approximation and a polynomial stepsize $k^{-ω}$, $ω\in (1/2,1)$. Assuming that the sequence of observed triples $(s_k,a_k,s_{k+1})_{k \geq 0}$ forms a uniformly geometrically ergodic Markov chain, and under suitable regularity conditions for the projected soft Bellman equation, we establish a Gaussian approximation bound in the convex distance with rate of order $n^{-1/4}$, up to polylogarithmic factors in $n$, where $n$ is the number of samples used by the algorithm. To obtain this result, we combine a linearization of the soft Bellman recursion with a Gaussian approximation for the leading martingale term. Finally, we derive high-order moment bounds for the algorithm's last iterate, which might be of independent interest.

Dimensionality reduction algorithms map high-dimensional data into visualizable 2D or 3D spaces, but traditionally rely on a discrete point-cloud paradigm. This discrete abstraction is susceptible to visual occlusion and artificial discontinuities, often failing to represent the continuous density of the underlying manifold. To address these limitations, we introduce Topo-GS, a framework that repurposes 3D Gaussian Splatting (3DGS) to cast multidimensional projection as a meshless volumetric reconstruction process. Instead of standard photometric losses, Topo-GS is driven by local geometric constraints. By solving orthogonal Procrustes targets, the optimization enforces an As-Rigid-As-Possible prior while explicitly aligning the spatial covariance of each Gaussian to the local tangent space. Recognizing that unrolling data of varying intrinsic dimensionalities requires distinct spatial treatments, we utilize a topology-aware strategy that tailors the loss formulation to preserve either continuous 1D trajectories or cohesive 2D surfaces. Quantitative and visual evaluations demonstrate that Topo-GS successfully transforms discrete scatter plots into continuous volumetric representations, where inherent projection distortions explicitly manifest as observable geometric variations, while preserving local topological fidelity comparable to discrete baselines.

A diffusion model trained on Hamiltonian trajectories can achieve rollout MSE near $10^{-3}$, but the standard deviation of its energy over time is between 7500 and 36000 times larger than the ground-truth energy standard deviation, indicating a failure to preserve conservation laws. This gap motivates our central question of whether neural networks can learn or select globally conserved quantities from physical trajectories. We investigate this across three Hamiltonian systems: projectile motion, pendulum, and spring-mass. We use a structured $T(v)+V(q)$ energy model, a black-box Conservation Discovery Network (CDN), a polynomial CDN, and a conditional diffusion baseline. The structured network reaches $R^2 \geq 0.9999$ against analytical energy on clean data, while the black-box CDN reaches $R^2 \geq 0.996$ when trained with temporal consistency plus a small alignment loss to analytical energy at $t=0$ ($λ_{\mathrm{align}}=0.2$). With $λ_{\mathrm{align}}=0$, CDN Pearson $R^2$ collapses on pendulum and spring-mass ($< 10^{-3}$), showing that temporal consistency alone is not enough to reliably identify the true energy. Under $1\%$ additive Gaussian noise, the CDN outperforms the structured model on the projectile and spring-mass systems, suggesting that the CDN may be more robust to noisy inputs in this setting. However, the polynomial CDN is sensitive to training configuration: it achieves $R^2=0.78$ under a short training schedule on the pendulum system, but reaches $R^2=0.9998$ with more training time and data, regardless of whether noise is added.

We study the query complexity of obtaining a relative Fisher information guarantee for sampling from a log-smooth non-log-concave distribution; this is a sampling analog of finding an approximate stationary point in optimization. Our algorithm is based on the proximal sampler, which is an implicit discretization of the Langevin diffusion, and requires an implementation of the backward step known as the restricted Gaussian oracle (RGO). We show that by leveraging the recent results for log-concave sampling with high-accuracy guarantees in Rényi divergence, we can obtain an approximate RGO implementation that -- when used with the proximal sampler -- yields a complexity guarantee in relative Fisher information that inherits the same dimension dependence as log-concave sampling, and improves upon prior work for non-log-concave sampling. We also show a converse reduction that any improvement in the dimension dependence in relative Fisher information for non-log-concave sampling will yield an improved dimension dependence for high-accuracy log-concave sampling.

Uncertainty quantification is essential in safety-critical settings--from autonomous driving to aviation, finance, and health--where decisions must rely on conservative bounds rather than point estimates. Predictor-level intervals (e.g., from quantile regression, conformal prediction, variance networks, or Bayesian models) generally do not compose: adding two per-variable intervals need not yield a valid interval for their sum or preserve coverage. In aviation, Gaussian overbounding replaces complex error distributions with a conservative Gaussian whose tails dominate the truth, so conservatism propagates through linear operations. Yet classical overbounds are global, often overly conservative, and hard to adapt to feature-conditioned errors. We propose a unified learning framework that trains neural networks to produce context-aware Gaussian overbounds--mean and scale--with provable conservatism on a finite quantile grid and, under three explicit regularity assumptions, continuous-tail conservatism on a certified interval. Our overbounding loss enforces conservativeness at selected quantiles while penalizing distributional distance with a Wasserstein-style term. The learned bounds support conservative linear-combination and convolution analysis on the enforced grid, and on the certified interval when assumptions hold, while being less redundant than traditional methods. We provide a scoped analysis of discrete-to-continuous conservatism and compact-domain objective regularity, and validate on synthetic data and real-world datasets, including multipath, ionospheric, and tropospheric residual errors. Across these settings, the method yields tighter bounds while maintaining conservatism on the enforced grid and in experiments. The framework is modality-agnostic and applicable to learning systems that require conservative, feature-conditioned uncertainty estimates in dynamic environments.

Reconstructing continuous flow fields from sparse surface-mounted sensors is central to aerodynamic design, flow control, and digital-twin instrumentation. Existing neural methods for this task typically encode sensor readings into implicit latent codes with little spatial interpretability and limited formal guidance on how representational capacity should scale with observation count. Inspired by 3D Gaussian Splatting, we introduce FLUIDSPLAT, a sensor-conditioned model that predicts K anisotropic Gaussian primitives forming a partition-of-unity scaffold, a spatially explicit and interpretable intermediate representation of the flow. For an idealized Gaussian primitive estimator, we prove an $O(K^{-s/d})$ approximation rate for fields with Sobolev smoothness $s$; incorporating $N$ noisy observations yields a squared-risk decomposition with bias $O(K^{-2s/d})$ and variance $O(σ^{2}K/N)$.Balancing the two yields $K^{*}\!\sim\!(N/σ^{2})^{d/(2s+d)}$: primitive count cannot grow freely under sparse sensing, revealing a variance bottleneck that motivates complementing the scaffold with a state-conditioned residual decoder. Across four benchmarks spanning 2D and 3D, FLUIDSPLAT achieves 11-28% error reduction over several strong baselines on cylinder flow, AirfRANS, FlowBench LDC-3D, and PhySense-Car 3D benchmarks.

Referring expression counting (REC) is an intention-driven task that requires context-aware visual reasoning. While recent vision-language models incorporate language for visual understanding, most existing REC methods rely on rulebased reinforcement learning with rewards focused primarily on final accuracy, overlooking the quality of intermediate reasoning. We propose REC-RL, a reinforcement learning framework that introduces a think-range-answer paradigm to explicitly optimize the visual reasoning process. RECRL employs Group Relative Policy Optimization and two lightweight rewards: an accuracy reward that combines range-based interval supervision with Gaussian-based precision guidance, and a format reward that enforces structured outputs. By modeling intermediate focus prediction as internal decision-making, REC-RL avoids additional annotations and better aligns with human perception. Extensive experiments demonstrate consistent improvements over strong baselines and robust generalization across benchmarks.

3D Gaussian Splatting (3DGS) has emerged as a leading representation for real-time novel view synthesis and has been widely adopted in various downstream applications. The core strength of 3DGS lies in its efficient kernel-based scene representation, where Gaussian primitives provide favorable mathematical and computational properties. However, under a finite primitive budget, the symmetric shape of each primitive directly affects representation compactness, especially near asymmetric structures such as object boundaries and one-sided surfaces. Recent works have explored more complex kernel distributions; however, they either remain within the elliptical family or rely on hard truncation, which limits continuous shape control and introduces distributional discontinuities. In this paper, we propose Skew-Normal Splatting (SNS), which adopts the Azzalini Skew-Normal distribution as the fundamental primitive. By introducing a learnable and bounded skewness parameter, SNS can continuously interpolate between symmetric Gaussians and Half-Gaussian-like shapes, enabling flexible modeling of both sharp boundaries and interior regions. Moreover, SNS preserves analytical tractability under affine transformations and marginalization. This property allows seamless integration into existing Gaussian Splatting rasterization pipelines. Furthermore, to address the strong coupling between scale, rotation, and skewness parameters, we introduce a decoupled parameterization and a block-wise optimization strategy to enhance training stability and accuracy. Extensive experiments on standard novel-view synthesis benchmarks show that SNS consistently improves reconstruction quality over Gaussian and recent non-Gaussian kernels, with clearer benefits on sharp boundaries and thin or one-sided structures.

Recent advances in 3D Gaussian Splatting (3DGS) have achieved remarkable success in high-fidelity Novel View Synthesis (NVS), yet the optimization process inevitably introduces noisy Gaussian primitives due to the sparse and incomplete initialization from Structure-from-Motion (SfM) point clouds. Most existing methods focus solely on adjusting the positions of primitives during optimization, while neglecting the underlying spatial structure. To this end, we introduce a new perspective by formulating the optimization of 3DGS as a primitive denoising process and propose Denoising-GS, a spatial-aware denoising framework for Gaussian primitives by taking both the positions and spatial structure into consideration. Specifically, we design an optimizer that preserves the spatial optimization flow of primitives, facilitating coherent and directed denoising rather than random perturbations. Building upon this, the Spatial Gradient-based Denoising strategy jointly considers the spatial supports of primitives to ensure gradient-consistent updates. Furthermore, the Uncertainty-based Denoising module estimates primitive-wise uncertainty to prune redundant or noisy primitives, while the Spatial Coherence Refinement strategy selectively splits primitives in sparse regions to maintain structural completeness. Experiments conducted on three benchmark datasets demonstrate that Denoising-GS consistently enhances NVS fidelity while maintaining representation compactness, achieving state-of-the-art performance across all benchmarks. Source code and models will be made publicly available.

Robust whole-slide image (WSI) analysis under strict data-governance remains challenging due to substantial cross-institutional stain heterogeneity. Domain generalization (DG) mitigates these shifts but typically requires centralized data, conflicting with privacy regulations. Federated learning (FedL) provides a decentralized alternative; however, existing FedL and federated DG (FedDG) approaches rely almost exclusively on low-order statistics, assuming Gaussian-like stain distributions. In contrast, real-world staining processes often produce asymmetric, heavy-tailed color distributions due to biochemical diffusion and scanner nonlinearity. Consequently, current methods fail to model the higher-order, non-Gaussian characteristics dominating real-world stain variability. To address this, we propose FedStain, a stain-aware FedDG framework explicitly incorporating higher-order stain moments--skewness and kurtosis--as compact statistical descriptors exchanged during federated optimization. These descriptors require no pixel-level data transmission, preserving strict privacy and communication efficiency, while enabling the global model to capture stain variability missed by low-order statistics. FedStain also employs a contrastive, cross-site parameter aggregation strategy to promote stain-invariant representations without relaxing data constraints. Extensive experiments on Camelyon17 and our new MvMidog-Fed benchmark show FedStain yields consistent improvements, outperforming state-of-the-art FedL, DG, and FedDG baselines by up to +3.9% absolute accuracy. To our knowledge, FedStain is the first FedDG approach to explicitly model higher-order stain statistics, enabling robust cross-institutional deployment in computational pathology.

We present BlitzGS, a distributed 3DGS framework that reduces active Gaussian workload for fast city-scale reconstruction. BlitzGS manages this workload at three coupled levels. At the system level, the framework shards Gaussians across GPUs by index parity rather than spatial blocks. This approach mitigates the cross-block visibility redundancy inherent in spatial partitioning. Furthermore, it distributes each rendering step through a single cross-GPU exchange that routes projected Gaussians to their tile owners. At the model level, scheduled importance-scoring passes shrink the global Gaussian population. During these passes, the framework generates a per-Gaussian visibility weight to bias density-control updates toward contributing primitives and a per-view importance mask for the view-level renderer. At the view level, BlitzGS trims each camera's active set with a distance-based LOD gate to exclude excessively fine primitives for the current frustum and the importance-based culling mask to skip Gaussians with negligible cross-view contribution. On large-scale benchmarks, BlitzGS matches the rendering quality of recent large-scale baselines while delivering an order-of-magnitude speedup, training city-scale scenes in tens of minutes. Our code is available at https: //github.com/AkierRaee/BlitzGS.

3D Language Gaussian Splatting (3DLGS) augments 3D Gaussian Splatting with language-aligned visual features for open-vocabulary 3D scene understanding. A core challenge is efficiently associating high-dimensional vision-language embeddings with millions of 3D Gaussians while preserving efficient feature rendering for text-based querying. Existing methods either store dense features directly on Gaussians, causing high storage costs and slow rendering, or learn compact representations through expensive per-scene optimization with repeated feature rasterization. No existing method simultaneously achieves fast 3D semantic reconstruction, efficient storage, and fast rendering. We propose SCOUP (Sparse COde UPlifting), which addresses all three by decoupling language representation learning from 3D Gaussian optimization. Rather than working directly in 3D, we learn sparse codebook-based representations entirely using features associated with 2D image regions, associating each region with a sparse set of codebook coefficients. We then uplift these coefficients to 3D Gaussians with our weighted sparse aggregation using Gaussian-to-pixel associations, where each Gaussian accumulates coefficients over codebook atoms across views. Top-$K$ filtering then extracts the most dominant multi-view coefficients per Gaussian, enabling efficient storage and fast rendering. Our method achieves up to $400\times$ training speedup while being $3\times$ more memory efficient during training compared to the state-of-the-art in rendering speed. Across multiple benchmarks, SCOUP matches or outperforms existing methods in open-vocabulary querying accuracy.

Reliable autonomous driving relies on large-scale, well-labeled data and robust models. However, manual data collection is resource-intensive, and traditional simulation suffers from a persistent reality gap. While recent generative frameworks and radiance-field methods improve visual fidelity, they still struggle with temporal and spatial consistency and cannot ensure physics-aware behavior, limiting their applicability to driving scenario generation. To address these challenges, we propose Real2Sim, an unified framework that combines 4D Gaussian Splatting (4DGS) with a differentiable Material Point Method (MPM) solver. Real2Sim explicitly reconstructs dynamic driving scenes as temporally continuous Gaussian primitives, supports instance-level editing, and simulates realistic object-object and object-environment interactions. This framework enables physics-aware, high-fidelity synthesis of diverse, editable scenarios, including challenging corner cases such as collisions and post-impact trajectories. Experiments on the Waymo Open Dataset validate Real2Sim's capabilities in rendering, reconstruction, editing, and physics simulation, demonstrating its potential as a scalable tool for data generation in downstream tasks such as perception, tracking, trajectory prediction, and end-to-end policy learning.

Recent advances in 3D Gaussian Splatting (3DGS) have enabled significant progress in photorealistic novel view synthesis. However, traditional 3DGS relies on a slow, iterative optimization process, which limits its use in scenarios demanding real-time results. To overcome this bottleneck, recent feed-forward methods aim to predict Gaussian attributes directly from images, but they often struggle with the redundancy of Gaussian primitives and rendering quality. In this work, we introduce a transformer-based architecture specifically designed for feed-forward Gaussian Splatting. Our key insight is that spatial and semantic relationships among Gaussians can be effectively captured through a sparse attention mechanism, enabled by a Z-order strategy that organizes the unstructured Gaussian set into a spatially coherent sequence. Furthermore, we incorporate this Z-order strategy to adaptively suppress redundancy while preserving critical structural details. This allows the transformer to efficiently model context, compress Gaussian primitives, and predict Gaussian attributes in a single forward pass. Comprehensive experiments demonstrate that our method achieves fast and high-quality novel view synthesis with fewer Gaussian primitives.

3D Gaussian Splatting (3DGS) is becoming a practical representation for novel view synthesis, but its growing adoption, together with rapid advances in instruction-driven 3DGS editing, also exposes a dual copyright risk: once a 3DGS-based asset is released, it can be used without permission and manipulated through 3D editing. Existing protection methods address only one side of this problem. Watermarking can trace ownership after unauthorized use, but it cannot prevent malicious editing. Adversarial edit-deterrence methods can disrupt editing, but they do not provide evidence of ownership. To the best of our knowledge, we present the first unified protection framework for 3DGS that jointly optimizes ownership tracing and unauthorized editing deterrence. Our framework combines a scene-wide watermarking objective over all Gaussians with an adversarial objective for edit deterrence. The adversarial branch combines latent-anchor separation, denoising-trajectory diversion, and cross-attention diversion to divert the editing trajectory, while an update-saliency-motivated Gaussian selection strategy assigns stronger adversarial updates to mask-selected Gaussians, improving the balance among watermark recovery, edit deterrence, and rendering fidelity. Experiments on scenes from Mip-NeRF 360 and Instruct-NeRF2NeRF demonstrate that the proposed framework achieves a favorable balance among bit accuracy, edit deterrence, and rendering quality. These results suggest that practical copyright protection of 3DGS-based assets can be more effectively addressed by integrating ownership tracing and unauthorized editing deterrence into a single optimization framework.

We propose an algorithm to sample from composite log-concave distributions over $\mathbb{R}^d$, i.e., densities of the form $π\propto e^{-f-g}$, assuming access to gradient evaluations of $f$ and a restricted Gaussian oracle (RGO) for $g$. The latter requirement means that we can easily sample from the density $\text{RGO}_{g,h,y}(x) \propto \exp(-g(x) -\frac{1}{2h}||y-x||^2)$, which is the sampling analogue of the proximal operator for $g$. If $f + g$ is $α$-strongly convex and $f$ is $β$-smooth, our sampler achieves $\varepsilon$ error in total variation distance in $\widetilde{\mathcal O}(κ\sqrt d \log^4(1/\varepsilon))$ iterations where $κ:= β/α$, which matches prior state-of-the-art results for the case $g=0$. We further extend our results to cases where (1) $π$ is non-log-concave but satisfies a Poincaré or log-Sobolev inequality, and (2) $f$ is non-smooth but Lipschitz.

Landslide monitoring and simulation play an important role in urban safety assessment and disaster prevention. Existing landslide simulation pipelines typically rely on digital elevation model and mesh-based representations, which are suitable for geometric analysis, but often lack visual realism. This limitation reduces their effectiveness in interactive applications, hazard communication, and public education. In this paper, we propose a UAV-based scan-to-simulation framework that bridges photorealistic scene capture and physics-based landslide simulation through 3DGS. Specifically, our pipeline includes four stages: (1) UAV-based acquisition of slope imagery, (2) reconstruction of a low-anisotropy 3DGS scene representation, (3) volumetric conversion of the target simulation region by filling the interior of the surface-based model, and (4) integration with the Material Point Method (MPM) for landslide simulation. We validate the proposed framework on a real landslide site in Hong Kong that experienced a severe landslide event. The results show that our method supports both realistic visual reconstruction and effective simulation.

We consider the active learning problem where the goal is to learn an unknown function with low prediction error under an unknown Boltzmann distribution induced by the function itself. This self-induced weighting arises naturally in problems such as potential energy surface (PES) modeling in computational chemistry, yet poses unique challenges as the target distribution is unknown and its partition function is intractable. We propose \texttt{AB-SID-iVAR}, a Gaussian Process-based acquisition function that approximates the intractable Bayesian target distribution in closed form while avoiding partition function estimation, and is applicable to both discrete and continuous input domains. We also analyze a Thompson sampling alternative (\texttt{TS-SID-iVAR}) as a higher variance Monte Carlo variant. Despite the unknown target, under mild conditions, we establish that the terminal prediction error vanishes with high probability, and provide a tighter average-case guarantee. We demonstrate consistent improvements over existing approaches in this setting on synthetic benchmarks and real-world PES modeling and drug discovery tasks.

Learning a physical model from video data that can comprehend physical laws and predict the future trajectories of objects is a formidable challenge in artificial intelligence. Prior approaches either leverage various Partial Differential Equations (PDEs) as soft constraints in the form of PINN losses, or integrate physics simulators into neural networks; however, they often rely on strong priors or high-quality geometry reconstruction. In this paper, we propose CausalGS, a framework that learns the causal dynamics of complex dynamic 3D scenes solely from multi-view videos, while dispensing with the reliance on explicit priors. At its core is an inverse physics inference module that decouples the complex dynamics problem from the video into the joint inference of two factors: the initial velocity field representing the scene's kinematics, and the intrinsic material properties governing its dynamics. This inferred physical information is then utilized within a differentiable physics simulator to guide the learning process in a physics-regularized manner. Extensive experiments demonstrate that CausalGS surpasses the state-of-the-art on the highly challenging task of long-term future frame extrapolation, while also exhibiting advanced performance in novel view interpolation. Crucially, our work shows that, without any human annotation, the model is able to learn the complex interactions between multiple physical properties and understand the causal relationships driving the scene's dynamic evolution, solely from visual observations.

In this paper, we aim to jointly model the geometry, appearance, and physical information of 3D scenes solely from dynamic multi-view videos, without relying on any physical priors. Existing works typically employ physical losses merely as soft constraints or integrate physical simulations into neural networks; however, these approaches often fail to effectively learn complex motion physics. Although modeling velocity fields holds the potential to capture authentic physical information, due to the lack of appropriate physical constraints, current methods are unable to correctly learn the interaction mechanisms between rigid and non-rigid particles. To address this, we propose VeloGauss, designed to learn the physical properties of complex dynamic 3D scenes without physical priors. Our method learns the velocity field for each Gaussian particle by introducing a Physics Code and a Particle Dynamics System, and ultimately incorporates Global Physical Constraints to ensure the physical consistency of the scene. Extensive experiments on four public datasets demonstrate that our method outperforms achieves state-of-the-art performance in both Novel View Interpolation and Future Frame Extrapolation tasks.

Point clouds are a fundamental representation for robotic perception tasks such as localization, mapping, and object pose estimation. However, LiDAR-acquired point clouds are inherently sparse and non-uniform, providing incomplete observations of the underlying scene geometry. This makes reliable geometric reasoning challenging and degrades downstream perception performance. Existing approaches attempt to compensate for these limitations by estimating local geometry, but often rely on hand-crafted statistics or end-to-end supervised learning, which can suffer from limited scalability or require large amounts of accurately labeled data. To address these challenges, we explicitly model point cloud geometry under a principled mathematical formulation. We represent local geometry as a statistical manifold induced by a family of Gaussian distributions, where each point is associated with a Gaussian capturing its local geometric structure. Based on this formulation, we introduce Point-to-Ellipsoid (POLI), a deep neural estimator that predicts per-point Gaussian geometry. POLI learns a mapping from point cloud observations to their underlying geometry in a self-supervised manner, removing the need for labeled data while preserving strong geometric inductive biases. The resulting representation integrates seamlessly into existing robotic perception pipelines without architectural modifications. Extensive experiments show that POLI enables accurate and robust geometry estimation and consistently improves performance across diverse robotic perception tasks.

While novel view synthesis (NVS) for dynamic scenes has seen significant progress, reconstructing temporally consistent geometric surfaces remains a challenge. Neural Radiance Fields (NeRF) and 3D Gaussian Splatting (3DGS) offer powerful dynamic scene rendering capabilities; however, relying solely on photometric optimization often leads to geometric ambiguities. This results in discontinuous surfaces, severe artifacts, and broken surfaces over time. To address these limitations, we present DySurface, a novel framework that bridges the effectiveness of explicit Gaussians with the geometric fidelity of implicit Signed Distance Functions (SDFs) in dynamic scenes. Our approach tackles the structural discrepancy between the forward deformation of 3DGS ($canonical \rightarrow dynamic$) and the backward deformation required for volumetric SDF rendering ($dynamic \rightarrow canonical$). Specifically, we propose the VoxGS-DSDF branch that leverages deformed Gaussians to construct a dynamic sparse voxel grid, providing explicit geometric guidance to the implicit SDF field. This explicit anchoring effectively regularizes the volumetric rendering process, significantly improving surface reconstruction quality, with watertight boundaries and detailed representations. Quantitative and qualitative experiments demonstrate that DySurface significantly outperforms state-of-the-art baselines in geometric accuracy metrics while maintaining competitive rendering performance.

This paper aims at analyzing the regularization effect that data augmentation induces on supervised regression methods in the proportional regime, where the number of covariates grows proportionally to the number of samples. We provide a tight characterization of the test error, measured in mean squared error, in terms only of the population quantities of the true data, as well as first and second order statistics of the augmentation scheme. Our results are valid under misspecified feature maps, and for any network architecture where only the last readout layer is trained, and the rest of the network is either frozen or randomly initialized. We specify our results in the case of Gaussian data, and show that our asymptotic characterization is tight in this setting.

High-quality, real-time talking head synthesis remains a fundamental challenge in computer vision. Existing reconstruction- and rendering-based methods typically rely on identity-specific models, limiting cross-identity generalization. To address this issue, we propose SDTalk, a one-shot 3D Gaussian Splatting (3DGS)-based framework that generalizes to unseen identities without personalized training or fine-tuning. Our framework comprises two modules with a two-stage training strategy. In the first stage, we incorporate structured facial priors into the reconstruction module and separately predict 3DGS parameters for visible and occluded regions, enabling complete head reconstruction from a single image. In the second stage, we introduce a dual-branch motion field to model coarse and fine facial dynamics, improving detail fidelity and lip synchronization. Experiments demonstrate that SDTalk surpasses existing methods in both visual quality and inference efficiency.

Most Gaussian Splatting techniques that provide a 3D semantic representation of the scene do not optimize the underlying 3D geometry, making object-level editing or asset extraction challenging. Recent methods, such as COBGS, Trace3D, ObjectGS, acknowledge this limitation and propose approaches that modify the scene's geometry to represent the underlying semantics. We advance this concept further by proposing a novel solution that provides near perfect boundaries in object extraction. We do so by introducing two new losses in the optimization that take care of: 1) a loss that modifies the geometry of visible Gaussians to respect semantic boundaries, and 2) a loss that adjusts the geometry of non-visible Gaussians that appear once the object is extracted. Our first loss propagates gradients directly through the rasterization, allowing for seamless integration within the optimization of the Gaussian parameters. The second loss also propagates gradients to Gaussian parameters but does so without passing through the rasterization, enabling modification of the scene's geometry even when little transmittance reaches a Gaussian (partial or non-visible). Exhaustive comparisons with 12 state of the art methods across 4 datasets, using six metrics, demonstrate that our approach produces overall the best boundary segmentation to date.

Accurate High-Definition (HD) map construction is critical for autonomous driving, yet existing methods face a fundamental trade-off: vectorization-based approaches preserve topology but struggle with geometric fidelity, while rasterization-based approaches enable precise geometric supervision but produce unstructured outputs. To bridge this gap, we propose GSMap, a novel framework that unifies both paradigms via a learnable 2D Gaussian representation. Each map element is modeled as an ordered sequence of 2D Gaussians, whose centers correspond to the vertices of the vectorized polyline/polygon. This formulation enables simultaneous optimization through: (1) Differentiable rasterization that enforces pixel-level geometric constraints, and (2) Topology-aware vectorization that maintains structural regularity. Experiments on both nuScenes and Argoverse2 demonstrate that our Gaussian-based representation effectively unifies geometric and topological learning, achieving significant performance improvements and demonstrating strong compatibility with existing HD mapping architectures. Code will be available at https://github.com/peakpang/GSMap

Reconstructing 3D fluid velocity fields from sparse 2D video observations is a highly ill-posed inverse problem, demanding both transport consistency with observed motion and physical validity under fluid laws. Existing methods typically impose these constraints through soft penalties, often leading to compromised accuracy and convergence issues. We introduce a reconstruction framework that structurally enforces both constraints. Specifically, we parameterize the reconstructed velocity using a continuous Divergence-Free Kernel representation, driving the advection of a Lagrangian 3D Gaussian Splatting representation. This formulation intrinsically guarantees both flow incompressibility and long-range transport coherence by construction. To enable the efficient optimization of such a constrained system, we introduce a novel Sliding Window scheme that propagates gradients over meaningful temporal horizons while maintaining tractable training costs. Experiments on synthetic and real-world datasets demonstrate that our method outperforms state-of-the-art baselines in both transport consistency and physical accuracy, enabling applications such as high-quality re-simulation and flow analysis.

Continuous-time state estimation is gaining in popularity due to its abilities to provide smooth solutions, handle asynchronous sensors, and interpolate between data points. While there are two main paradigms, parametric (e.g., temporal basis functions, splines) and nonparametric (Gaussian processes), the latter has seen less adoption despite its technical advantages and relative ease of implementation. In this article, we seek to rectify this situation by providing a new simplified explanation of GP continuous-time estimation rooted in the language of factor graphs, which have become the de facto estimation paradigm in much of robotics. To simplify onboarding, we also provide three working examples implemented in the popular GTSAM estimation framework.

Virtual production (VP) use LED walls to provide both background imagery and image-based lighting. While this enables on-set compositing, it couples lighting to background and scene appearance, limiting flexibility for downstream editing. In addition, inverse rendering conventionally relies on physically-based rendering to estimates 3D geometry and lighting, using environment maps. However, these maps are typically low-resolution and assume far-field lighting. In VP, with near-field and high-resolution image-based lighting, this can lead to inaccuracies and introduce complexities when editing. Addressing this, we propose a VP-specific framework for 3D reconstruction and relighting using Gaussian Splatting. This uses the known background imagery to condition the relighting process. This avoids relying on environment maps and reduces compositing to a background-image editing task. To realize our framework, we introduce a process (and associated dataset) that captures real VP scenes under varying background content and illumination conditions. This data is used to decompose a 3D scene into fixed appearance and variable lighting components. The variable lighting process simulates light transport by parameterizing each primitive with a UV coordinate, intensity value and resolution modifier. Using mipmaps, these directly sample the background texture in image space - implicitly capturing reflections and refractions without physically-based rendering. Combined with the fixed appearance component, this allows us to render relit scenes using a Gaussian Splatting rasterizer. Compared to baselines, our approach achieves higher-quality 3D reconstruction and controllable relighting. The method is efficient (<3 GB RAM, <5 GB VRAM, <2 hours training, ~35 FPS) and supports rendering useful arbitrary output variables including depth, lighting intensity, lighting color, and unlit renders.

$L_1$-Approximating polynomials, i.e., polynomials that approximate indicator functions in $L_1$-norm under certain distributions, are widely used in computational learning theory. We study the existence of \textit{non-negative} $L_1$-approximating polynomials with respect to Gaussian distributions. This is a stronger requirement than $L_1$-approximation but weaker than sandwiching polynomials (which themselves have many applications). These non-negative approximating polynomials have recently found uses in smoothed learning from positive-only examples. In this short note, we prove that every class of sets with Gaussian surface area (GSA) at most $Γ$ under the standard Gaussian admits degree-$k$ non-negative polynomials that $\eps$-approximate its indicator functions in $L_1$-norm, for $k=\tilde{O}(Γ^2/\varepsilon^2)$. Equivalently, finite GSA implies $L_1$-approximation with the stronger pointwise guarantee that the approximating polynomial has range contained in $[0,\infty)$. Up to a constant-factor, this matches the degree of the best currently known Gaussian $L_1$-approximation degree bound without the non-negativity constraint.

We present Density-Sampled Gaussians (DeG), a novel 3D representation designed to bridge the gap between adaptive rendering primitives and scalable generative modeling. Unlike existing approaches that constrain 3D Gaussians to fixed voxel grids or arrays, DeG models Gaussian centers as samples from a learnable probability density function defined over an octree. This formulation provides a rigorous mathematical framework for adaptive density control: by jointly optimizing the spatial density and Gaussian attributes under rendering supervision, our model naturally concentrates primitives in regions of high geometric complexity. We achieve this via a new render loss contribution gradient that serves as a fully differentiable analogue to the discrete densification and pruning heuristics used in standard Gaussian Splatting. The resulting representation is highly flexible, supporting variable-resolution decoding from a single latent code by simply adjusting the sampling budget. To enable generative synthesis, we train a latent diffusion model on DeG. We identify a critical challenge in applying diffusion to unordered set-structured latents, which can significantly slow convergence, and propose VecSeq, a canonical re-indexing mechanism that anchors latent tokens to a deterministic 3D Sobol sequence. This transforms the ambiguous set-generation problem into a robust sequence modeling task. Extensive experiments demonstrate that our pipeline achieves state-of-the-art quality in single-image-to-3D generation, combining the structural adaptivity of unstructured primitives with the training stability of grid-based methods.

We introduce CUTS-GPR, a new method for performing numerically exact Gaussian process regression (GPR) in high-dimensional settings. The key component of CUTS-GPR is an extremely fast kernel matrix-vector product, which exhibits near-linear or even linear scaling with the amount of training data, $N$, and low-order polynomial scaling with dimensionality, $D$. This is obtained by combining an additive kernel with an incomplete grid and exploiting the resulting structure of the kernel matrix. We demonstrate the scalability of the matrix-vector product by running benchmarks with billions of data points and thousands of dimensions. Full GPR calculations, including hyperparameter optimization, are completed in a matter of hours for $N = 447 265$ and $D = 24$. We demonstrate that our CUTS-GPR enables Bayesian modeling of high-dimensional potential energy surfaces - a longstanding challenge in computational chemistry.

Explicit neural representations such as 3D Gaussian Splatting (3DGS) enable high-fidelity and real-time novel view synthesis, yet optimize for alpha-composited optical appearance rather than ray-intersectable geometry. In contrast, radio-frequency (RF) digital twins require deterministic multi-bounce paths, where the geometry dictates trajectories and their associated attenuation and delay. We introduce a framework enabling differentiable RF propagation simulation directly within visually reconstructed neural scenes, allowing point-to-point path computation between arbitrary 3D locations while preserving high-quality visual rendering. Unlike conventional RF simulation pipelines that rely on manually constructed meshes, we embed Gaussian primitives into a hardware-accelerated ray tracing structure as the underlying spatial representation. By extracting physically meaningful channel impulse responses from visual-only reconstructions, we provide cross-modal evidence that neural reconstructions can serve as unified spatial representations for both electromagnetic propagation simulation and photorealistic view synthesis.

We study Flow Matching in a semi-discrete setting where a Gaussian source is transported toward a discrete target supported on finitely many points. This semi-discrete regime is the theoretical setting behind the use of Flow Matching for generative modeling, where the target distribution is represented by a finite dataset. In this semi-discrete regime, the exact Flow Matching velocity field is available in closed form, which makes it possible to analyze the geometry induced by the terminal flow map independently of optimization and approximation effects. We investigate the terminal assignment regions, namely the preimages of the target atoms under the terminal flow. We show that these regions are open, simply connected and, under an additional assumption, homeomorphic to the unit ball. At the same time, a planar four-point example shows that these cells can differ sharply from Laguerre cells arising in semi-discrete optimal transport: they may be non-convex, have curved boundaries, and exhibit different boundedness and adjacency patterns. These results clarify the geometry intrinsically induced by the exact semi-discrete Flow Matching objective before neural approximation enters the picture.

While Gaussian Splatting-based Feature Fields (GSFFs) have shown promise for visual localization, this paper highlights that photometrically optimized GSFFs are inherently ill-suited for 2D-3D matching. The volumetric extent of each Gaussian induces many-to-one pixel-to-point mappings that destabilize PnP-based pose estimation, while photometric optimization gives rise to superfluous Gaussians devoid of multi-view consistency. To address these issues, we propose SplitGS-Loc, a localization-specialized GSFFs construction framework that disambiguates 2D-3D correspondences by exploiting Gaussian attributes. Our key design, Mixture-of-Gaussians-based splitting, decomposes each Gaussian into smaller Gaussians, replacing ambiguous many-to-one with precise one-to-one correspondences. In parallel, we exploit composition weights from GS rasterization to select Gaussians that significantly and consistently contribute across multiple views and aggregate discriminative features through strong pixel-Gaussian associations, enforcing multi-view consistency. The resulting compact yet discriminative feature fields enable stable PnP convergence, achieving state-of-the-art performance on localization benchmarks. Extensive experiments validate that SplitGS-Loc extends the utility of photometric GSFFs to accurate and efficient localization by exploiting Gaussian attributes, without per-scene training or iterative pose refinement.

Generalizable novel view synthesis aims to render unseen views from uncalibrated input images without requiring per-scene optimization. Recent feed-forward approaches based on 3D Gaussian Splatting have achieved promising efficiency and rendering quality. However, most of them assign a fixed number of Gaussians to each pixel or voxel, ignoring the spatially varying complexity of real-world scenes. Such uniform allocation often wastes Gaussian primitives in smooth regions while providing insufficient capacity for fine structures, complex geometry, and high-frequency details. This motivates us to predict region-dependent primitive cardinalities rather than impose a fixed primitive budget everywhere, enabling a more expressive 3D scene representation. Therefore, we propose SplatWeaver, a generalizable novel view synthesis framework that is able to dynamically allocate Gaussian primitives over different regions in a feed-forward manner. Specifically, SplatWeaver introduces cardinality Gaussian experts and a pixel-level routing scheme, wherein each expert specializes in producing a specific number of primitives from 0 to M, and the routing scheme coordinates these experts to adaptively determine how many Gaussian primitives should be allocated to each spatial location. Moreover, SplatWeaver incorporates a high-frequency prior with attendant guidance module and routing regularization to stabilize expert selection and promote complexity-aware allocation. By leveraging high-frequency cues, the routing process is encouraged to assign more Gaussian primitives to fine structures and textured regions, while suppressing redundancy in smooth areas. Extensive experiments across diverse scenarios show that SplatWeaver consistently outperforms state-of-the-art methods, delivering more faithful novel-view renderings with fewer Gaussian primitives. Project Page: https://yecongwan.github.io/SplatWeaver/

Sparse-view satellite image surface reconstruction remains highly challenging, fundamentally because the reliability of multi-view matching under satellite imaging conditions is strongly spatially heterogeneous. Affected by large photometric differences, weak textures, and repetitive textures, multi-view geometric constraints are often sparse, unevenly distributed, and locally unreliable. Although 2D Gaussian Splatting (2DGS) is more suitable than 3D Gaussian Splatting (3DGS) for the explicit representation of continuous surfaces, research on generalizable feed-forward 2DGS frameworks for sparse-view satellite surface reconstruction is still lacking. To address this issue, we propose SatSurfGS, a generalizable sparse-view surface reconstruction method for satellite imagery based on 2DGS. The proposed method builds a coarse-to-fine Gaussian attribute prediction framework and explicitly models local geometric reliability at three levels: feature learning, Gaussian parameter estimation, and training optimization. Specifically, we propose a confidence-aware monocular multi-view feature fusion module to adaptively integrate monocular priors and multi-view matching features according to local confidence; a cross-stage self-consistency residual guidance module to stabilize stage-wise Gaussian parameter refinement using the residual between the rendered height map from the previous stage and the current-stage MVS height map, together with confidence information; and a confidence bidirectional routing loss to achieve differentiated allocation of geometric and appearance supervision. Experiments on satellite datasets show that the proposed method achieves improved rendering quality, surface reconstruction accuracy, cross-dataset generalization, and inference efficiency compared with representative generalizable baselines and competitive per-scene optimization methods.

Edge learning refers to training machine learning models deployed on edge platforms, typically using new data accumulated onboard. The computational limitations on edge devices affect not only model optimisation, but also calculation of the predictive uncertainty of the current model on the unlabelled data, which is vital for informing model updating. In this paper, we investigate edge learning in the context of performing deep image regression on a remote sensing satellite, where a deep network is executed by an onboard computer to regress a scalar $y$ from an input image, e.g., $y$ is the percentage of pixels indicating cloud coverage or land use. We propose an uncertainty-guided edge learning (UGEL) algorithm that can accurately prioritise the data to speed up training convergence of the on-board regression model. Underpinning UGEL is the calculation of predictive uncertainty based on deep beta regression, where a deep network is used to estimate the parameters of a beta distribution for which the target $y$ for an input image has a high likelihood. Compared to established methods for uncertainty estimation that are either too costly on edge devices (e.g., require many forward passes per sample) or make strict assumptions on the predictive distribution (e.g., Gaussian), deep beta regression is computable in a single forward pass and allows more general predictive distributions. Results show that UGEL delivers faster-converging edge learning than active or semi-supervised learning. Code and models are publicly available at https://github.com/anh-vunguyen/UGEL.

In this note, we report five mathematical discoveries made in collaboration with Grok, all of which have been subsequently verified by the authors. These include an improved lower bound on the maximal Gaussian perimeter of convex sets in $\mathbb{R}^n$, sharper $L_2$-$L_1$ moment comparison inequalities on the Hamming cube $\{-1,1\}^n$, a strengthened autoconvolution inequality, improved asymptotic bounds on the size of the largest $g$-Sidon sets in $\{1,\dots,n\}$, and an optimal balanced Szarek's inequality.