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February 2019 Statistics on the Stiefel manifold: Theory and applications
Rudrasis Chakraborty, Baba C. Vemuri
Ann. Statist. 47(1): 415-438 (February 2019). DOI: 10.1214/18-AOS1692

Abstract

A Stiefel manifold of the compact type is often encountered in many fields of engineering including, signal and image processing, machine learning, numerical optimization and others. The Stiefel manifold is a Riemannian homogeneous space but not a symmetric space. In previous work, researchers have defined probability distributions on symmetric spaces and performed statistical analysis of data residing in these spaces. In this paper, we present original work involving definition of Gaussian distributions on a homogeneous space and show that the maximum-likelihood estimate of the location parameter of a Gaussian distribution on the homogeneous space yields the Fréchet mean (FM) of the samples drawn from this distribution. Further, we present an algorithm to sample from the Gaussian distribution on the Stiefel manifold and recursively compute the FM of these samples. We also prove the weak consistency of this recursive FM estimator. Several synthetic and real data experiments are then presented, demonstrating the superior computational performance of this estimator over the gradient descent based nonrecursive counter part as well as the stochastic gradient descent based method prevalent in literature.

Citation

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Rudrasis Chakraborty. Baba C. Vemuri. "Statistics on the Stiefel manifold: Theory and applications." Ann. Statist. 47 (1) 415 - 438, February 2019. https://doi.org/10.1214/18-AOS1692

Information

Received: 1 February 2017; Revised: 1 February 2018; Published: February 2019
First available in Project Euclid: 30 November 2018

zbMATH: 07036206
MathSciNet: MR3909937
Digital Object Identifier: 10.1214/18-AOS1692

Subjects:
Primary: 62F12
Secondary: ‎58A99

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 1 • February 2019
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