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.Comment: 10 pages, 4 figures; final versio

On the Effective Measure of Dimension in the Analysis Cosparse Model

Many applications have benefited remarkably from low-dimensional models in the recent decade. The fact that many signals, though high dimensional, are intrinsically low dimensional has given the possibility to recover them stably from a relatively small number of their measurements. For example, in compressed sensing with the standard (synthesis) sparsity prior and in matrix completion, the number of measurements needed is proportional (up to a logarithmic factor) to the signal's manifold dimension. Recently, a new natural low-dimensional signal model has been proposed: the cosparse analysis prior. In the noiseless case, it is possible to recover signals from this model, using a combinatorial search, from a number of measurements proportional to the signal's manifold dimension. However, if we ask for stability to noise or an efficient (polynomial complexity) solver, all the existing results demand a number of measurements which is far removed from the manifold dimension, sometimes far greater. Thus, it is natural to ask whether this gap is a deficiency of the theory and the solvers, or if there exists a real barrier in recovering the cosparse signals by relying only on their manifold dimension. Is there an algorithm which, in the presence of noise, can accurately recover a cosparse signal from a number of measurements proportional to the manifold dimension? In this work, we prove that there is no such algorithm. Further, we show through numerical simulations that even in the noiseless case convex relaxations fail when the number of measurements is comparable to the manifold dimension. This gives a practical counter-example to the growing literature on compressed acquisition of signals based on manifold dimension.Comment: 19 pages, 6 figure

On the Power of Adaptivity in Matrix Completion and Approximation

We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence assumptions on the matrix row space that are necessary for passive sampling procedures. For exact recovery of a low-rank matrix, our algorithm judiciously selects a few columns to observe in full and, with few additional measurements, projects the remaining columns onto their span. This algorithm exactly recovers an $n \times n$ rank $r$ matrix using $O(nr\mu_0 \log^2(r))$ observations, where $\mu_0$ is a coherence parameter on the column space of the matrix. In addition to completely eliminating any row space assumptions that have pervaded the literature, this algorithm enjoys a better sample complexity than any existing matrix completion algorithm. To certify that this improvement is due to adaptive sampling, we establish that row space coherence is necessary for passive sampling algorithms to achieve non-trivial sample complexity bounds. For constructing a low-rank approximation to a high-rank input matrix, we propose a simple algorithm that thresholds the singular values of a zero-filled version of the input matrix. The algorithm computes an approximation that is nearly as good as the best rank-$r$ approximation using $O(nr\mu \log^2(n))$ samples, where $\mu$ is a slightly different coherence parameter on the matrix columns. Again we eliminate assumptions on the row space