Sparse Representation of White Gaussian Noise with Application to L0-Norm Decoding in Noisy Compressed Sensing

TL;DR

This paper derives the limits and distribution of sparse representations of white Gaussian noise using an overcomplete dictionary, introducing a sharp threshold for 0-norm decoding in noisy compressed sensing with implications for mean-square error analysis.

Contribution

It provides the first derivation of achievable and converse regions for sparse Gaussian noise representation and introduces a sharp 0-norm decoding threshold in noisy compressed sensing.

Findings

Derived achievable and converse regions for sparse noise representation.

Inferred the marginal distribution of sparse representations.

Introduced a sharp threshold for 0-norm decoding in noisy compressed sensing.

Abstract

The achievable and converse regions for sparse representation of white Gaussian noise based on an overcomplete dictionary are derived in the limit of large systems. Furthermore, the marginal distribution of such sparse representations is also inferred. The results are obtained via the Replica method which stems from statistical mechanics. A direct outcome of these results is the introduction of sharp threshold for $\ell_{0}$-norm decoding in noisy compressed sensing, and its mean-square error for underdetermined Gaussian vector channels.