probabilistic geometry densification

This work addresses the limitations of conventional Gaussian approximations and particle filters in Bayesian inference for non-Gaussian, multimodal posterior distributions. We propose a geometric Bayesian inference framework based on ensembles of uniform distributions over convex polytopes. Our method models the posterior as a recursively updated collection of convex polytopes, each supporting a uniform distribution—marking the first such formulation. Crucially, we introduce a Kalman-style polytope update mechanism that integrates ensemble filtering principles with convex geometric operations, including Ikeda mappings and Minkowski sums. Unlike moment-based or point-estimate approaches, our method inherently captures support-set structure and uncertainty propagation without relying on statistical moments. Experiments on the low-dimensional Ikeda system and high-dimensional Lorenz’96 system demonstrate substantial improvements in modeling accuracy and numerical stability for non-Gaussian posteriors, thereby extending the expressive capacity beyond classical approximation methods.

Markov chain Monte Carlo (MCMC) methods suffer from low sampling efficiency, high sensitivity to initial values, and insufficient theoretical guarantees when targeting high-dimensional heavy-tailed distributions. Method: This paper proposes three adaptive spherical MCMC algorithms—spherical random walk (SRW), spherical slice sampler (SSS), and spherical bouncy particle sampler (SBPS)—unifying spherical stereographic projection with online parameter adaptation for the first time. Contribution/Results: We establish the first ergodicity analysis framework for continuous-time adaptive MCMC, rigorously proving L² convergence, almost-sure convergence, and a central limit theorem. Empirically, the algorithms exhibit robustness and accelerated convergence on non-centered, inhomogeneous targets; they remain stable even when initialized far from the mode, achieving significantly higher sampling efficiency than state-of-the-art methods.

Bayesian posterior sampling on Riemannian manifolds—such as Stiefel and Grassmann manifolds—is challenging, especially under anisotropic target densities; existing gradient-based or preconditioning-dependent methods suffer from poor robustness. Method: We propose the first geodesic-based slice sampling MCMC algorithm for Riemannian manifolds, generalizing Euclidean hit-and-run slice sampling by replacing straight-line segments with geodesics. Our method is gradient-free, requires no pre-tuning, satisfies detailed balance, and is geometrically adaptive. Contribution/Results: We establish theoretical guarantees of ergodicity and detailed balance. Empirical evaluation on synthetic and real-world data demonstrates faster convergence and superior mixing compared to state-of-the-art manifold MCMC methods. Crucially, our algorithm remains stable and efficient even under highly anisotropic posteriors, significantly enhancing the practicality and robustness of Bayesian inference on matrix manifolds.

Poisson surface reconstruction from partial or sequential point clouds using Gaussian processes (GPs) traditionally involves a two-stage pipeline—first interpolating point-wise normals via GP regression, then solving the volumetric Poisson PDE globally—entailing high computational cost and reliance on diagonal approximations of the kernel matrix inverse. Method: We propose a unified, single-stage framework that embeds geometric Gaussian processes directly into the Poisson reconstruction formulation, jointly modeling surface geometry and uncertainty. This integrates normal interpolation and PDE solving into one sparse linear system, enabling mesh-free function evaluation and local spatial reasoning—including collision detection, on-demand ray casting, and slice-level view planning—without explicit kernel matrix operations. Results: Experiments demonstrate superior reconstruction accuracy and real-time performance over conventional two-stage approaches, validating its efficacy for task-driven, online 3D perception.

This work addresses the challenge of accurately quantifying uncertainty in existing semi-dense matching methods, which often overlook catastrophic failures during the coarse matching stage, leading to biased geometric estimates. To remedy this, the authors propose a lightweight post-processing framework that explicitly incorporates coarse matching failures into uncertainty modeling. Specifically, they introduce a two-component calibrated Laplacian mixture model with only nine learnable parameters to capture the long-tailed distribution arising from both local refinement noise and coarse matching outliers. Additionally, a CoRe (Coarse-and-Refined) geometric refitting module is designed to leverage Bayesian posterior probabilities for generating soft correspondence weights, enabling reweighted optimization. The method consistently improves downstream geometric accuracy across diverse pretrained matchers and robust estimators while incurring minimal computational overhead.

This work addresses the problem of exact sampling from the distribution over partial triangulations of a convex polygon, where each triangulation is weighted proportionally to an exponential function of its number of diagonals. The authors propose the first direct randomized sampling algorithm that does not rely on Markov chains, leveraging combinatorial structure analysis and geometric probability modeling to achieve exact sampling from the target distribution. This approach overcomes the efficiency limitations inherent in traditional Markov chain methods, achieving an expected time complexity of $O((n\sqrt{\lambda} + 1)\log n)$, which significantly outperforms existing techniques and demonstrates near-optimal performance on large-scale instances.

This work addresses the insufficient integration of learning-based methods and geometric constraints in camera pose and scene structure estimation by proposing a modular framework. The approach first employs a learning model (VGGT) to generate initial hypotheses for depth and relative pose, which are subsequently refined and validated using classical geometric algorithms such as point-to-plane RGB-D ICP. Crucially, the framework explicitly distinguishes the roles of learning as a “proposer” and geometry as a “referee,” emphasizing that the geometric module serves not merely as post-processing but as an essential mechanism for verifying and integrating learned outputs. Experiments on the TUM RGB-D dataset demonstrate that, in moderately challenging rigid scenes, the system significantly outperforms both purely learning-based and purely geometric baselines when the learned depth aligns geometrically with the camera intrinsics and undergoes optimization by the geometric backend.

This work addresses the limitations of traditional Laplace approximations, which fail to capture posterior skewness, heavy tails, and narrow high-probability regions, as well as the computational expense of existing wrapped Gaussian methods that require evaluating geodesics, Christoffel symbols, or curvature tensors. By leveraging contrast function theory on a statistical manifold equipped with the Fisher–Rao metric and prior-induced geometry, the authors derive, for the first time, closed-form approximations of the exponential and logarithmic maps. This enables an efficient wrapped Gaussian approximation that avoids costly geometric computations. The proposed method substantially reduces computational complexity while accurately capturing complex posterior geometries across diverse models, achieving speedups of several orders of magnitude over current state-of-the-art approaches.

This work addresses the limitation of traditional probabilistic circuits, which employ data-agnostic mixture weights and thus struggle to capture the local geometric structure of data manifolds. To explicitly incorporate geometric information, we introduce Voronoi tessellation into the sum nodes of probabilistic circuits—a novel approach in this domain. We propose a differentiable relaxation mechanism to enable gradient-based learning, design an approximate inference method with provable error bounds, and derive structural constraints under which exact inference can be recovered. Experimental results on standard density estimation benchmarks demonstrate that our method effectively balances expressive geometric modeling with tractable inference.