Shannon Theoretic Limits on Noisy Compressive Sampling

TL;DR

This paper establishes fundamental limits on the number of noisy measurements needed to recover sparse signals in complex space, showing linear measurement bounds are optimal for certain regimes, improving upon previous logarithmic bounds.

Contribution

It proves that linear measurement complexity is both necessary and sufficient for sparse recovery in noisy settings when sparsity grows linearly with ambient dimension, surpassing prior convex programming bounds.

Findings

Linear measurements are sufficient for sparse recovery with noise.

Linear measurements are necessary when sparsity scales linearly with dimension.

Logarithmic measurement bounds are required in the sublinear sparsity regime.

Abstract

In this paper, we study the number of measurements required to recover a sparse signal in ${\mathbb C}^M$ with $L$ non-zero coefficients from compressed samples in the presence of noise. For a number of different recovery criteria, we prove that $O(L)$ (an asymptotically linear multiple of $L$) measurements are necessary and sufficient if $L$ grows linearly as a function of $M$. This improves on the existing literature that is mostly focused on variants of a specific recovery algorithm based on convex programming, for which $O(L\log(M-L))$ measurements are required. We also show that $O(L\log(M-L))$ measurements are required in the sublinear regime ($L = o(M)$).