Minimax rank estimation for subspace tracking

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.

💡 Research Summary

The paper tackles the classic model‑order selection problem of determining the rank (or dimensionality) of a signal subspace in array processing, a task that is surprisingly under‑addressed despite its importance in many statistical signal‑processing applications. The authors work within the standard linear observation model y(t)=As(t)+w(t), where A∈ℂ^{N×r} spans the true signal subspace, s(t)∈ℂ^{r} are the source signals, and w(t) is additive white Gaussian noise with variance σ². From T snapshots they form the sample covariance matrix (\hat R = \frac{1}{T}\sum_{t=1}^{T} y(t) y(t)^{H}).

Using recent results from random matrix theory, they first derive the asymptotic eigenvalue distribution of (\hat R) in the high‑dimensional regime (N,T→∞ with N/T→c∈(0,∞)). The Marčenko–Pastur law describes the bulk of noise eigenvalues, while the Baik–Ben Arous–Péché (BBP) phase transition tells us that a population eigenvalue λ_i (i.e., a signal power) will generate a “spiked” sample eigenvalue that separates from the bulk only if λ_i exceeds a critical threshold
\