Sparse Signal Separation in Redundant Dictionaries

TL;DR

This paper introduces a unified framework for separating signals sparse in different redundant dictionaries, providing new guarantees and comparisons between analysis and synthesis methods using coherence-based metrics.

Contribution

It develops a hybrid approach that unifies analysis and synthesis sparse signal separation, offering novel recovery guarantees and a basis for performance comparison.

Findings

Recovery guarantees for the proposed separation algorithm.

Unification of analysis and synthesis approaches.

Comparison with existing D-RIP based results.

Abstract

We formulate a unified framework for the separation of signals that are sparse in "morphologically" different redundant dictionaries. This formulation incorporates the so-called "analysis" and "synthesis" approaches as special cases and contains novel hybrid setups. We find corresponding coherence-based recovery guarantees for an l1-norm based separation algorithm. Our results recover those reported in Studer and Baraniuk, ACHA, submitted, for the synthesis setting, provide new recovery guarantees for the analysis setting, and form a basis for comparing performance in the analysis and synthesis settings. As an aside our findings complement the D-RIP recovery results reported in Cand\`es et al., ACHA, 2011, for the "analysis" signal recovery problem: minimize_x ||{\Psi}x||_1 subject to ||y - Ax||_2 \leq {\epsilon}, by delivering corresponding coherence-based recovery results.