Phase Transitions for Support Recovery from Gaussian Linear Measurements

TL;DR

This paper investigates the support recovery problem from Gaussian linear measurements, revealing a phase transition in sample complexity around the ratio of support size to measurements per sample, and proposes an optimal algorithm for both regimes.

Contribution

It introduces a phase transition analysis for support recovery with Gaussian measurements and provides an optimal algorithm for measurement-constrained regimes.

Findings

Sample complexity exhibits a phase transition at k/m=1.

Proposed algorithm is optimal in both measurement regimes.

Multiple measurements per sample are more valuable when m << k.

Abstract

We study the problem of recovering the common $k$-sized support of a set of $n$ samples of dimension $d$, using $m$ noisy linear measurements per sample. Most prior work has focused on the case when $m$ exceeds $k$, in which case $n$ of the order $(k/m)\log(d/k)$ is both necessary and sufficient. Thus, in this regime, only the total number of measurements across the samples matter, and there is not much benefit in getting more than $k$ measurements per sample. In the measurement-constrained regime where we have access to fewer than $k$ measurements per sample, we show an upper bound of $O((k^{2}/m^{2})\log d)$ on the sample complexity for successful support recovery when $m\ge 2\log d$. Along with the lower bound from our previous work, this shows a phase transition for the sample complexity of this problem around $k/m=1$. In fact, our proposed algorithm is sample-optimal in both the regimes. It follows that, in the $m\ll k$ regime, multiple measurements from the same sample are more valuable than measurements from different samples.