Improved Sparsity Thresholds Through Dictionary Splitting

TL;DR

This paper introduces improved sparsity thresholds for compressed sensing by analyzing dictionaries as concatenations of sub-dictionaries, resulting in more accurate thresholds based on coherence parameters.

Contribution

It provides new sparsity thresholds that are explicitly derived from the coherence of sub-dictionaries, enhancing the understanding of dictionary structures in compressed sensing.

Findings

Better sparsity thresholds for union of sub-dictionaries

Thresholds explicitly depend on coherence parameters

Improved recovery guarantees over traditional coherence-based bounds

Abstract

Known sparsity thresholds for basis pursuit to deliver the maximally sparse solution of the compressed sensing recovery problem typically depend on the dictionary's coherence. While the coherence is easy to compute, it can lead to rather pessimistic thresholds as it captures only limited information about the dictionary. In this paper, we show that viewing the dictionary as the concatenation of two general sub-dictionaries leads to provably better sparsity thresholds--that are explicit in the coherence parameters of the dictionary and of the individual sub-dictionaries. Equivalently, our results can be interpreted as sparsity thresholds for dictionaries that are unions of two general (i.e., not necessarily orthonormal) sub-dictionaries.