"Compressed" Compressed Sensing

TL;DR

This paper investigates whether prior knowledge about a vector's distribution can reduce the number of samples needed for compressed sensing, providing bounds and showing cases where reduction is or isn't possible.

Contribution

It derives information-theoretic bounds and proposes a simple thresholding estimator to analyze sample complexity with prior knowledge.

Findings

Sample reduction possible for discrete or highly distorted vectors

In many cases, no reduction in sample size is achievable

Bounds connect compressed sensing with free probability theory

Abstract

The field of compressed sensing has shown that a sparse but otherwise arbitrary vector can be recovered exactly from a small number of randomly constructed linear projections (or samples). The question addressed in this paper is whether an even smaller number of samples is sufficient when there exists prior knowledge about the distribution of the unknown vector, or when only partial recovery is needed. An information-theoretic lower bound with connections to free probability theory and an upper bound corresponding to a computationally simple thresholding estimator are derived. It is shown that in certain cases (e.g. discrete valued vectors or large distortions) the number of samples can be decreased. Interestingly though, it is also shown that in many cases no reduction is possible.