Minimax rank estimation for subspace tracking

TL;DR

This paper introduces a new, asymptotically minimax optimal method for estimating the rank of a signal subspace in noisy array data, utilizing random matrix theory and phase transition analysis.

Contribution

It develops a parameter-free, threshold-based rank estimation method with proven asymptotic minimax optimality, advancing subspace tracking techniques.

Findings

Identifies a phase transition threshold for eigenvalue informativeness.

Proposes a simple, asymptotically optimal rank selection rule.

Demonstrates practical efficacy through simulation studies.

Abstract

Rank estimation is a classical model order selection problem that arises in a variety of important statistical signal and array processing systems, yet is addressed relatively infrequently in the extant literature. Here we present sample covariance asymptotics stemming from random matrix theory, and bring them to bear on the problem of optimal rank estimation in the context of the standard array observation model with additive white Gaussian noise. The most significant of these results demonstrates the existence of a phase transition threshold, below which eigenvalues and associated eigenvectors of the sample covariance fail to provide any information on population eigenvalues. We then develop a decision-theoretic rank estimation framework that leads to a simple ordered selection rule based on thresholding; in contrast to competing approaches, however, it admits asymptotic minimax optimality and is free of tuning parameters. We analyze the asymptotic performance of our rank selection procedure and conclude with a brief simulation study demonstrating its practical efficacy in the context of subspace tracking.