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方向与射影形状AI 解读

This paper deals with projective shape analysis, which is a study of finite configurations of points modulo projective transformations. The topic has various applications in machine vision. We introduce a convenient projective shape space, as well as an appropriate coordinate system for this shape space. For generic configurations of k points in m dimensions, the resulting projective shape space is identified as a product of k−m−2 copies of axial spaces ℝPm. This…

流形上内蕴与外蕴样本均值的大样本理论（二）AI 解读

This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Fréchet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on Riemannian manifolds. Central limit theorems are also obtained for extrinsic sample means w.r.t. an arbitrary embedding of a differentiable manifold in a Euclidean space. Bootstrap methods particularly suitable for these…

流形上内蕴与外蕴样本均值的大样本理论AI 解读

Sufficient conditions are given for the uniqueness of intrinsic and extrinsic means as measures of location of probability measures Q on Riemannian manifolds. It is shown that, when uniquely defined, these are estimated consistently by the corresponding indices of the empirical $\hat Q_n$. Asymptotic distributions of extrinsic sample means are derived. Explicit computations of these indices of $\hat Q_n$ and their asymptotic dispersions are carried out for distributions on…