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再生核希尔伯特空间在函数型分类中的应用AI 解读

The Hájek–Feldman dichotomy establishes that two Gaussian measures are either mutually absolutely continuous with respect to each other (and hence there is a Radon–Nikodym density for each measure with respect to the other one) or mutually singular. Unlike the case of finite-dimensional Gaussian measures, there are nontrivial examples of both situations when dealing with Gaussian stochastic processes. This article provides: (a) Explicit expressions for the optimal (Bayes)…

庞加莱锥性质及其统计应用AI 解读

A domain $S\subset\mathbb{R}^{d}$ is said to fulfill the Poincaré cone property if any point in the boundary of $S$ is the vertex of a (finite) cone which does not otherwise intersects the closure $\bar{S}$. For more than a century, this condition has played a relevant role in the theory of partial differential equations, as a shape assumption aimed to ensure the existence of a solution for the classical Dirichlet problem on $S$. In a completely different setting, this paper…

长度与表面积估计的非参数方法AI 解读

The Minkowski content L0(G) of a body G⊂ℝd represents the boundary length (for d=2) or the surface area (for d=3) of G. A method for estimating L0(G) is proposed. It relies on a nonparametric estimator based on the information provided by a random sample (taken on a rectangle containing G) in which we are able to identify whether every point is inside or outside G. Some theoretical properties concerning strong consistency, L1-error and convergence rates are obtained. A…

支撑集估计的插入式方法AI 解读

We suggest a new approach, based on the use of density estimators, for the problem of estimating the (compact) support of a multivariate density. This subject (motivated in terms of pattern analysis by Grenander) has interesting connections with detection and clustering. A natural class of density-based estimators is defined. Universal consistency results and convergence rates are established for these estimators, with respect to the usual measure-based metric $d_{\mu}$ betwe…