wrapped gaussian approximation

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 expressive power of geometric Gaussian approximations—specifically, pushing a standard Gaussian distribution onto complex target distributions (e.g., Bayesian posteriors) via diffeomorphisms or Riemannian exponential maps—and investigates whether a single diffeomorphism can uniformly approximate an entire family of distributions with high fidelity. Method: We establish rigorous theoretical guarantees for geometric Gaussian approximation, introducing a unified approximation framework for distribution families and proving equivalence between diffeomorphism-based and Riemannian exponential-map-based constructions. Results: We prove universality: any continuous probability distribution can be approximated to arbitrary precision by a Gaussian pushed forward via some diffeomorphism. Moreover, we construct an explicit, family-wide uniform approximation scheme and demonstrate formal equivalence between the two geometric approximation paradigms. These results provide a solid theoretical foundation and a novel design paradigm for efficient, interpretable probabilistic modeling in Bayesian inference.

Bayesian posteriors are often skewed, whereas mainstream deterministic approximations—such as Laplace’s method and variational Bayes—rely on symmetric densities (e.g., Gaussians), leading to systematic bias and reduced accuracy. Method: We propose a generic, optimization-free skewness-aware perturbation framework that can be seamlessly integrated with any off-the-shelf symmetric approximation. Our approach constructs analytical perturbations based on skew-symmetric density families, unifying asymptotic expansion and variational analysis. Contribution/Results: We theoretically establish finite-sample accuracy improvement and prove that the asymptotic convergence rate is accelerated by at least a factor of √n. The method is model-agnostic and compatible with diverse symmetric approximation paradigms. Numerical experiments demonstrate substantial gains over standard Gaussian approximations—particularly in moderate-to-small sample regimes and under strong posterior skewness—empirically validating the predicted convergence acceleration and robustness.

This work addresses the bias and over-concentration of classical Laplace approximation in Bayesian inference—particularly under complex models and limited data—where the Gaussian posterior approximation becomes excessively narrow and inaccurate. We propose a Riemannian-geometric improvement centered on the Fisher information metric, redesigning the curvature-aware metric structure to yield, for the first time, an asymptotically unbiased and exact Laplace approximation in the infinite-data limit. Building on this foundation, we introduce two novel variants that systematically correct biases arising from suboptimal metric choices in existing Riemannian Laplace methods. Our theoretical analysis extends the asymptotic statistical framework, establishing rigorous conditions for consistency and accuracy. Empirical evaluation demonstrates substantial improvements in posterior approximation fidelity and calibration, with robust performance even in finite-sample regimes.

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.

Efficient sampling from multimodal, unnormalized density functions on Riemannian manifolds remains challenging, as existing methods often fail to simultaneously respect the underlying geometric structure and capture complex distributional characteristics. This work proposes a training-free sampling framework that extends the principles of diffusion models to Riemannian manifolds for the first time. By constructing geometrically compatible stochastic interpolation paths and coupling them with nonequilibrium deterministic dynamics, the method gradually transports an easily sampled noise distribution toward the target distribution. Relying solely on standard Monte Carlo techniques and incorporating iterative posterior sampling, the approach demonstrates strong empirical performance in high-dimensional, heavy-tailed, and multimodal settings, offering both theoretical rigor and broad applicability.

This work investigates the stability of Gaussian inference on smooth manifolds, where marginalization and conditioning typically yield non-Gaussian distributions influenced by underlying geometry, complicating the assessment of linearization-based methods. Focusing on tangent-space linearization, the study establishes the first explicit non-asymptotic Wasserstein-2 (W₂) stability bound, cleanly separating local second-order geometric distortion from non-local tail leakage effects. The proposed closed-form diagnostic depends only on the mean, covariance, and proxies for curvature or injectivity radius, revealing that normal-direction uncertainty dominates error when locality assumptions break down. Experiments on toroidal and planar systems demonstrate a sharp degradation in calibration performance when √|Σ|_op/R ≈ 1/6, providing a practical trigger for switching to multi-chart or sampling-based manifold inference schemes.

本文解决了计算高斯核距离昂贵的问题，通过使用随机傅里叶特征(RFF)方法，在保持相对误差的前提下，有效降低了计算成本。

This work addresses the challenge of applying Gaussian processes (GPs) to high-dimensional inputs, where they are prone to the curse of dimensionality, and existing two-stage dimensionality reduction approaches often compromise either predictive accuracy or reliable uncertainty quantification. To overcome this limitation, the authors propose an end-to-end Bayesian joint modeling framework that seamlessly integrates input dimensionality reduction within the GP formulation. By placing a prior on the Stiefel manifold to enforce orthogonality of the projection matrix and employing Riemannian Hamiltonian Monte Carlo for posterior inference along geodesics, the method achieves, for the first time, a unified Bayesian treatment of GP regression and dimensionality reduction. The framework is further extended to deep Gaussian processes to enhance representational capacity. Experimental results demonstrate superior performance over conventional two-stage methods in both prediction accuracy and uncertainty calibration, albeit at increased computational cost.

本文针对高斯核密度统计问题，利用其几何和解析特性，提出了一种单遍次线性空间近似算法，有效处理了流数据中的相似性感知统计。