Convergence of the reach for a sequence of Gaussian-embedded manifolds
Convergence of the reach for a sequence of Gaussian-embedded manifolds
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DOI：
10.1007/s00440-017-0801-1
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发表时间：
2015-03
影响因子：
2
通讯作者：
R. Adler;Sunder Ram Krishnan;Jonathan E. Taylor;S. Weinberger
中科院分区：
文献类型：
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作者：
R. Adler;Sunder Ram Krishnan;Jonathan E. Taylor;S. Weinberger
受流形学习问题的启发，我们研究了一系列随机流形，这些随机流形是通过通过一系列高斯映射将固定的、紧凑的流形嵌入到维度递增的欧几里得球体而生成的。流形学习定理的基本平滑参数之一是 M 的范围或临界半径。粗略地说，范围是流形偏离凸性的度量，它结合了局部曲率和全局拓扑。本文发展了一系列随机高斯嵌入流形范围的极限理论，为全局范围建立了几乎肯定的收敛性，并为其及其局部版本建立了涨落理论。全局范围收敛到一个在高斯过程的再生核希尔伯特空间理论及其极值理论中众所周知的常数。
Motivated by questions of manifold learning, we study a sequence of random manifolds, generated by embedding a fixed, compact manifoldMinto Euclidean spheres of increasing dimension via a sequence of Gaussian mappings. One of the fundamental smoothness parameters of manifold learning theorems is the reach, or critical radius, ofM. Roughly speaking, the reach is a measure of a manifold’s departure from convexity, which incorporates both local curvature and global topology. This paper develops limit theory for the reach of a family of random, Gaussian-embedded, manifolds, establishing both almost sure convergence for the global reach, and a fluctuation theory for both it and its local version. The global reach converges to a constant well known both in the reproducing kernel Hilbert space theory of Gaussian processes, as well as in their extremal theory.