Estimation of Sparsity via Simple Measurements

TL;DR

This paper investigates minimal measurement strategies to estimate the sparsity of a vector using various operations, unifying multiple problems like group testing and compressed sensing through coding theory and probabilistic methods.

Contribution

It provides tight bounds on the number of measurements needed for sparsity estimation across different operational models, generalizing existing frameworks.

Findings

O(D log D log n) measurements suffice for OR-based sparsity estimation.

Nearly this many measurements are necessary for OR-based case.

Linear and finite field multiplication cases require Θ(D) and Θ(D log_q(n/D)) measurements respectively.

Abstract

We consider several related problems of estimating the 'sparsity' or number of nonzero elements $d$ in a length $n$ vector $\mathbf{x}$ by observing only $\mathbf{b} = M \odot \mathbf{x}$, where $M$ is a predesigned test matrix independent of $\mathbf{x}$, and the operation $\odot$ varies between problems. We aim to provide a $\Delta$-approximation of sparsity for some constant $\Delta$ with a minimal number of measurements (rows of $M$). This framework generalizes multiple problems, such as estimation of sparsity in group testing and compressed sensing. We use techniques from coding theory as well as probabilistic methods to show that $O(D \log D \log n)$ rows are sufficient when the operation $\odot$ is logical OR (i.e., group testing), and nearly this many are necessary, where $D$ is a known upper bound on $d$. When instead the operation $\odot$ is multiplication over $\mathbb{R}$ or a finite field $\mathbb{F}_q$, we show that respectively $\Theta(D)$ and $\Theta(D \log_q \frac{n}{D})$ measurements are necessary and sufficient.