Nullspace Property for Optimality of Minimum Frame Angle Under Invertible Linear Operators

TL;DR

This paper investigates how preconditioning linear operators can improve sparse signal recovery by reducing frame coherence and enhancing conditioning, supported by theoretical analysis and numerical simulations.

Contribution

It formulates a convex optimization approach to design preconditioners that lower coherence and improve frame conditioning for better sparse recovery guarantees.

Findings

Preconditioners can significantly reduce frame coherence.

Improved conditioning correlates with better sparse recovery.

Numerical results validate the theoretical benefits of the proposed method.

Abstract

Orthogonal Matching Pursuit and Basis Pursuit are popular reconstruction algorithms for recovery of sparse signals. The exact recovery property of both the methods has a relation with the coherence of the underlying redundant dictionary, i.e. a frame. A frame with low coherence provides better guarantees for exact recovery. An equivalent formulation of the associated linear system is obtained via premultiplication by a non-singular matrix. In view of bounds that guarantee sparse recovery, it is very useful to generate the preconditioner in such way that the preconditioned frame has low coherence as compared to the original. In this paper, we discuss the impact of preconditioning on sparse recovery. Further, we formulate a convex optimization problem for designing the preconditioner that yields a frame with improved coherence. In addition to reducing coherence, we focus on designing well conditioned frames and numerically study the relationship between the condition number of the preconditioner and the coherence of the new frame. Alongside theoretical justifications, we demonstrate through simulations the efficacy of the preconditioner in reducing coherence as well as recovering sparse signals.