Minimax rank estimation for subspace tracking

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.

10 pages, 4 figures; final version

References in corpus (3)

• Bi-cross-validation of the SVD and the nonnegative matrix factorization
• Central limit theorems for eigenvalues in a spiked population model
• Accuracy of the Tracy-Widom limit for the largest eigenvalue in white Wishart matrices

Cited by in corpus (8)

• Asymptotic power of sphericity tests for high-dimensional data
• Cross-Validation for Unsupervised Learning
• Learning the nonlinear geometry of high-dimensional data: Models and algorithms
• Fast estimation of approximate matrix ranks using spectral densities
• Spherical Cap Packing Asymptotics and Rank-Extreme Detection
• Source Number Estimation via Entropy Estimation of Eigenvalues (EEE) in Gaussian and Non-Gaussian Noise
• Optimal Covariance Estimation for Condition Number Loss in the Spiked Model
• Rank-Extreme Association of Gaussian Vectors and Low-Rank Detection