The Annals of Statistics

Statistics on the Stiefel manifold: Theory and applications

Rudrasis Chakraborty and Baba C. Vemuri

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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.

Article information

Source
Ann. Statist., Volume 47, Number 1 (2019), 415-438.

Dates
Received: February 2017
Revised: February 2018
First available in Project Euclid: 30 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1543568593

Digital Object Identifier
doi:10.1214/18-AOS1692

Mathematical Reviews number (MathSciNet)
MR3909937

Zentralblatt MATH identifier
07036206

Subjects
Primary: 62F12: Asymptotic properties of estimators
Secondary: 58A99: None of the above, but in this section

Keywords
Homogeneous space Stiefel manifold Fréchet mean Gaussian distribution

Citation

Chakraborty, Rudrasis; Vemuri, Baba C. Statistics on the Stiefel manifold: Theory and applications. Ann. Statist. 47 (2019), no. 1, 415--438. doi:10.1214/18-AOS1692. https://projecteuclid.org/euclid.aos/1543568593


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