gaussian integral analysis

This work addresses the efficient computation of exponential expectation integrals of the form $ \mathbb{E} \exp\left\{\sum_{i=1}^m \phi_i\right\} $ under standard Gaussian or symmetric exponential measures in high-dimensional spaces, where each function $ \phi_i $ depends only on a small subset of coordinates. By leveraging the local dependency structure of the functions together with their Lipschitz constants, the study establishes, for the first time, verifiable sufficient conditions ensuring that the integral remains bounded away from zero. The approach integrates probabilistic measure analysis, Lipschitz estimates, and dependency graph modeling to provide a novel theoretical foundation for the non-degeneracy of high-dimensional integrals. This framework is successfully applied to problems in high-dimensional volume estimation and lattice point counting in polytopes, significantly enhancing computational feasibility.

Gaussian processes (GPs) struggle to rigorously incorporate uncountably infinite-dimensional functional prior information—such as boundary conditions or global physical constraints satisfied by PDE solutions. Method: This paper proposes a unified modeling framework grounded in reproducing kernel Hilbert spaces (RKHS), establishing for the first time a rigorous equivalence between the GP conditional expectation and orthogonal projection in RKHS. This enables direct embedding of functional constraints (e.g., Dirichlet or Neumann boundary conditions) into the GP prior, bypassing conventional pseudo-point approximations. Contribution/Results: We provide theoretical guarantees on existence, uniqueness, and convergence of the constrained GP posterior. Computationally, we design a practical numerical approximation algorithm. Experiments on PDE inverse problems demonstrate substantial improvements in uncertainty quantification accuracy and posterior consistency. The framework delivers a rigorous, general, and computationally tractable paradigm for integrating domain knowledge into Bayesian modeling.

This work addresses the fundamental bottleneck in Bayesian quadrature and MMD-based inference—namely, the lack of closed-form kernel mean embeddings (KMEs)—which severely limits the practical applicability of kernel methods. We introduce the first systematic, scalable dictionary of closed-form KMEs. Our method establishes a unified derivation framework grounded in kernel algebra and probability distribution transformations, integrating symbolic computation with probabilistic integral transforms to automatically generate novel analytical KME solutions. The dictionary covers数十 combinations of widely used kernels—including RBF, Matérn, and periodic kernels—and distributions—such as Gaussian, Gamma, Beta, and mixtures—with rigorous mathematical derivations and empirical validation. To facilitate adoption, we release *kme*, a lightweight, open-source Python library. This tool substantially lowers implementation barriers for kernel methods in numerical integration, statistical hypothesis testing, and Bayesian inference, thereby bridging the gap between theoretical kernel statistics and real-world applications.

This work addresses intrinsic dimension estimation using Gaussian kernels. We establish the first finite-sample concentration and anti-concentration inequalities with explicit dependence on key parameters—sample size, bandwidth, local manifold curvature, and density regularity—quantifying their precise impact on estimation error. We propose a novel adaptive bandwidth selection heuristic leveraging density derivative information, overcoming limitations of conventional empirical rules; theoretically, it mitigates the bias–variance trade-off, and numerical experiments confirm substantial improvements in estimation stability and accuracy. Crucially, we are the first to rigorously characterize how regularity conditions—specifically, Lipschitz continuity of the density and bounded curvature—quantitatively constrain the statistical convergence rate. Our results provide both a rigorous theoretical foundation and practical methodology for intrinsic dimension inference in high-dimensional manifold learning.

This paper addresses the problem of efficiently approximating function integrals in a reproducing kernel Hilbert space (RKHS) given only i.i.d. samples from the target distribution. We propose a subsampling strategy based on (approximate) leverage scores—the first application of leverage scores to RKHS numerical integration—which drastically reduces the number of function evaluations required. Theoretically, we prove that only (m = O(log n)) subsampled points suffice to preserve the optimal (n^{-1/2}) convergence rate, and our error bound adapts to the smoothness of the integrand, achieving minimax-optimal rates in Sobolev spaces. Empirically, the method significantly improves the accuracy–efficiency trade-off over random or greedy quadrature on real-world datasets. Our results directly enable scalable computation of maximum mean discrepancy (MMD) and facilitate the design of efficient kernel-based hypothesis tests.

This work addresses kernel regression under non-Gaussian noise—including sub-Gaussian, bounded, sub-exponential, and moment-bounded types—by establishing a unified probabilistic uniform error bound within a non-asymptotic framework. It overcomes the prevailing limitation of existing approaches that assume conditionally independent sub-Gaussian noise, and for the first time simultaneously accommodates multiple non-Gaussian noise distributions and dependent noise structures. By integrating concentration inequalities with reproducing kernel Hilbert space theory, the derived non-conservative error bounds substantially enhance the reliability of uncertainty quantification. This leads to significantly tighter confidence regions in safety-critical control tasks, outperforming current state-of-the-art methods in both theoretical rigor and practical performance.

This work addresses the inefficiency of existing algorithms for kernel mean estimation in high-dimensional settings when the error tolerance ε is small and the data diameter Δ is moderate. To overcome this limitation, the authors propose a novel fast spherical embedding technique that preserves local Euclidean distances while effectively controlling the global diameter of the embedded data, thereby mitigating distance collapse in high dimensions. This approach achieves, for the first time, a joint optimization of local distance preservation and global diameter constraints. As a result, it establishes a new upper bound on the time complexity for kernel mean estimation of Õ(d + εΔ² + 1/ε³), which significantly improves upon prior bounds of O(d/ε²), Õ(d + 1/ε⁴), and Õ(d + Δ²/ε²) in regimes with small ε and moderate Δ.

本文扩展了有限维线性逆问题中最小均方误差估计器的形式至无限维情况，使用广义样条和广义高斯过程解决该类问题。

This study addresses the theoretical bottleneck wherein the relationships among expressiveness concepts—such as characteristicity and universality—remain unknown for unbounded kernels. By leveraging reproducing kernel Hilbert space (RKHS) theory and functional analysis techniques, we conduct rigorous mathematical derivations. As the first work to systematically elucidate the intrinsic connections among various kernel expressiveness notions in the unbounded setting, this project clarifies the statistical properties of unbounded kernels and their associated RKHSs under mild assumptions, thereby establishing fundamental theoretical links between key concepts. This research fills a critical gap in the expressiveness theory of unbounded kernels and strengthens the theoretical foundations of kernel methods.

This work addresses the challenge of jointly inferring multi-level functional states—such as curves, derivatives, and integrals—in functional data modeling, where existing methods struggle to account for derivative uncertainty, cross-level covariance, and identifiability of integration constants. The authors propose an anchored Gaussian process differential ensemble framework that explicitly models integration constants by embedding anchor points together with their mean-square derivatives and repeated integrals into a joint Gaussian state, enabling efficient computation via transformed Hilbert spaces. A key innovation is the separation of anchor-induced covariance from boundary uncertainty, revealing that integration constants cannot be uniquely identified from anchors alone. To enhance derivative recovery accuracy, the method introduces the TARTARE calibration strategy. Theoretical analysis combines Laplace–Dirichlet basis functions, finite-rank approximations, and operator-level approximation bounds. Experiments demonstrate substantially improved posterior derivative estimation in second-order simulations while preserving accuracy in anchors and integrals, and a motorcycle crash case study confirms coherent inference of coupled kinematic states and functional turning points.