New Guarantees for Blind Compressed Sensing

TL;DR

This paper establishes new theoretical guarantees for blind compressed sensing, especially for overcomplete dictionaries, by integrating sparse coding and low-rank matrix recovery theories, and proposes an efficient measurement scheme.

Contribution

It develops the first comprehensive theoretical bounds for unconstrained BCS, extending to overcomplete dictionaries and incorporating recent sparse coding and low-rank recovery results.

Findings

Provides perfect recovery bounds for unconstrained BCS

Introduces an efficient measurement scheme for practical BCS

Analyzes BCS performance with polynomial-time sparse coding algorithms

Abstract

Blind Compressed Sensing (BCS) is an extension of Compressed Sensing (CS) where the optimal sparsifying dictionary is assumed to be unknown and subject to estimation (in addition to the CS sparse coefficients). Since the emergence of BCS, dictionary learning, a.k.a. sparse coding, has been studied as a matrix factorization problem where its sample complexity, uniqueness and identifiability have been addressed thoroughly. However, in spite of the strong connections between BCS and sparse coding, recent results from the sparse coding problem area have not been exploited within the context of BCS. In particular, prior BCS efforts have focused on learning constrained and complete dictionaries that limit the scope and utility of these efforts. In this paper, we develop new theoretical bounds for perfect recovery for the general unconstrained BCS problem. These unconstrained BCS bounds cover the case of overcomplete dictionaries, and hence, they go well beyond the existing BCS theory. Our perfect recovery results integrate the combinatorial theories of sparse coding with some of the recent results from low-rank matrix recovery. In particular, we propose an efficient CS measurement scheme that results in practical recovery bounds for BCS. Moreover, we discuss the performance of BCS under polynomial-time sparse coding algorithms.