Typical $l_1$-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices

TL;DR

This paper analyzes the limits of $l_1$-recovery for sparse vectors using concatenated random orthogonal matrices, revealing conditions where recovery performance surpasses universal predictions, especially with non-uniform signal densities.

Contribution

It develops a theoretical framework using the replica method to evaluate $l_1$-recovery limits for concatenated orthogonal matrices, highlighting non-universality in recovery performance.

Findings

Concatenated orthogonal matrices can improve recovery performance.

Non-uniform non-zero signal densities lead to better recovery than universal predictions.

Numerical experiments confirm theoretical predictions.

Abstract

We consider the problem of recovering an $N$-dimensional sparse vector $\vm{x}$ from its linear transformation $\vm{y}=\vm{D} \vm{x}$ of $M(< N)$ dimension. Minimizing the $l_{1}$-norm of $\vm{x}$ under the constraint $\vm{y} = \vm{D} \vm{x}$ is a standard approach for the recovery problem, and earlier studies report that the critical condition for typically successful $l_1$-recovery is universal over a variety of randomly constructed matrices $\vm{D}$. For examining the extent of the universality, we focus on the case in which $\vm{D}$ is provided by concatenating $\nb=N/M$ matrices $\vm{O}_{1}, \vm{O}_{2},..., \vm{O}_\nb$ drawn uniformly according to the Haar measure on the $M \times M$ orthogonal matrices. By using the replica method in conjunction with the development of an integral formula for handling the random orthogonal matrices, we show that the concatenated matrices can result in better recovery performance than what the universality predicts when the density of non-zero signals is not uniform among the $\nb$ matrix modules. The universal condition is reproduced for the special case of uniform non-zero signal densities. Extensive numerical experiments support the theoretical predictions.