Where is Randomness Needed to Break the Square-Root Bottleneck?

TL;DR

This paper investigates the minimal randomness needed in compressed sensing to overcome the square-root bottleneck, focusing on the role of structured dictionaries and coherence properties.

Contribution

It extends previous results on orthonormal bases to more general dictionaries, identifying conditions under which randomness can be reduced to break the bottleneck.

Findings

Breaking the bottleneck depends on the coherence and size of a sub-dictionary.

Structured dictionaries with low coherence sub-dictionaries enable reduced randomness.

The results generalize the two-ONB case to broader dictionary classes.

Abstract

As shown by Tropp, 2008, for the concatenation of two orthonormal bases (ONBs), breaking the square-root bottleneck in compressed sensing does not require randomization over all the positions of the nonzero entries of the sparse coefficient vector. Rather the positions corresponding to one of the two ONBs can be chosen arbitrarily. The two-ONB structure is, however, restrictive and does not reveal the property that is responsible for allowing to break the bottleneck with reduced randomness. For general dictionaries we show that if a sub-dictionary with small enough coherence and large enough cardinality can be isolated, the bottleneck can be broken under the same probabilistic model on the sparse coefficient vector as in the two-ONB case.