On the Role of Diversity in Sparsity Estimation

TL;DR

This paper investigates how diversity in observations affects the estimation of sparsity patterns, deriving bounds that reveal tradeoffs between measurement accuracy, diversity, and uncertainty, and demonstrating that optimal diversity improves estimation performance.

Contribution

It introduces bounds on joint sparsity pattern estimation that incorporate diversity effects, revealing key tradeoffs and showing optimal diversity enhances estimation accuracy.

Findings

Diversity introduces a tradeoff between noise uncertainty and nonzero value uncertainty.

Optimal diversity significantly improves estimation performance.

Bounds improve upon existing results even without diversity.

Abstract

A major challenge in sparsity pattern estimation is that small modes are difficult to detect in the presence of noise. This problem is alleviated if one can observe samples from multiple realizations of the nonzero values for the same sparsity pattern. We will refer to this as "diversity". Diversity comes at a price, however, since each new realization adds new unknown nonzero values, thus increasing uncertainty. In this paper, upper and lower bounds on joint sparsity pattern estimation are derived. These bounds, which improve upon existing results even in the absence of diversity, illustrate key tradeoffs between the number of measurements, the accuracy of estimation, and the diversity. It is shown, for instance, that diversity introduces a tradeoff between the uncertainty in the noise and the uncertainty in the nonzero values. Moreover, it is shown that the optimal amount of diversity significantly improves the behavior of the estimation problem for both optimal and computationally efficient estimators.