Sample-Measurement Tradeoff in Support Recovery under a Subgaussian Prior

TL;DR

This paper investigates the support recovery problem in a multi-sample setting with subgaussian priors, revealing that fewer than k measurements per sample require significantly more samples, specifically on the order of (k^2/m^2) log k(d-k), for exact support recovery.

Contribution

It establishes the fundamental limits of support recovery with limited measurements per sample, showing that the total number of measurements must scale quadratically with k when m<k.

Findings

Support recovery requires Θ((k^2/m^2) log k(d-k)) samples.

Fewer than k measurements per sample necessitate more samples for exact support recovery.

Support recovery is impossible with fewer than k total measurements when m<k.

Abstract

Data samples from $\mathbb{R}^{d}$ with a common support of size $k$ are accessed through $m$ random linear projections (measurements) per sample. It is well-known that roughly $k$ measurements from a single sample are sufficient to recover the support. In the multiple sample setting, do $k$ overall measurements still suffice when only $m$ measurements per sample are allowed, with $m<k$? We answer this question in the negative by considering a generative model setting with independent samples drawn from a subgaussian prior. We show that $n=\Theta((k^2/m^2)\cdot\log k(d-k))$ samples are necessary and sufficient to recover the support exactly. In turn, this shows that when $m<k$, $k$ overall measurements are insufficient for support recovery; instead we need about $m$ measurements each from $k^{2}/m^2$ samples, i.e., $k^{2}/m$ overall measurements are necessary.