Geometrical and statistical properties of M-estimates of scatter on Grassmann manifolds

TL;DR

This paper studies robust M-estimates of scatter on Grassmann manifolds, utilizing geometric properties of the space to establish convergence and a central limit theorem for these estimators.

Contribution

It introduces a geometric approach to analyze M-estimates of scatter on Grassmann manifolds, proving convergence and asymptotic normality using CAT(0) space properties.

Findings

Almost sure convergence of estimators as sample size increases

Central limit theorem for rescaled estimators

Identification of the Grassmannian as a boundary of a CAT(0) space

Abstract

We consider data from the Grassmann manifold $G(m,r)$ of all vector subspaces of dimension $r$ of $\mathbb{R}^m$, and focus on the Grassmannian statistical model which is of common use in signal processing and statistics. Canonical Grassmannian distributions $\mathbb{G}_{\Sigma}$ on $G(m,r)$ are indexed by parameters $\Sigma$ from the manifold $\mathcal{M}= Pos_{sym}^{1}(m)$ of positive definite symmetric matrices of determinant $1$. Robust M-estimates of scatter (GE) for general probability measures $\mathcal{P}$ on $G(m,r)$ are studied. Such estimators are defined to be the maximizers of the Grassmannian log-likelihood $-\ell_{\mathcal{P}}(\Sigma)$ as function of $\Sigma$. One of the novel features of this work is a strong use of the fact that $\mathcal{M}$ is a CAT(0) space with known visual boundary at infinity $\partial \mathcal{M}$. We also recall that the sample space $G(m,r)$ is a part of $\partial \mathcal{M}$, show the distributions $\mathbb{G}_{\Sigma}$ are $SL(m,\mathbb{R})$--quasi-invariant, and that $\ell_{\mathcal{P}}(\Sigma)$ is a weighted Busemann function. Let $\mathcal{P}_n =(\delta_{U_1}+\cdots+\delta_{U_n})/n$ be the empirical probability measure for $n$-samples of random i.i.d. subspaces $U_i\in G(m,r)$ of common distribution $\mathcal{P}$, whose support spans $\mathbb{R}^m$. For $\Sigma_n$ and $\Sigma_{\mathcal{P}}$ the GEs of $\mathcal{P}_n$ and $\mathcal{P}$, we show the almost sure convergence of $\Sigma_n$ towards $\Sigma$ as $n\to\infty$ using methods from geometry, and provide a central limit theorem for the rescaled process $C_n = \frac{m}{tr(\Sigma_{\mathcal{P}}^{-1} \Sigma_n)}g^{-1} \Sigma_n g^{-1}$, where $\Sigma =gg$ with $g\in SL(m,\mathbb{R})$ the unique symmetric positive-definite square root of $\Sigma$.