Unlabeled Compressed Sensing from Multiple Measurement Vectors

TL;DR

This paper develops a Bayesian algorithm based on Bi-VAMP for unlabeled compressed sensing with multiple measurement vectors, capable of recovering structured signals from noisy, permuted observations, and demonstrates its superior performance through theoretical analysis and numerical experiments.

Contribution

It introduces a novel UCS algorithm that generalizes unlabeled sensing to handle large permutation matrices without partial assumptions, using a bilinear AMP framework.

Findings

The UCS algorithm achieves accurate signal recovery in noisy, permuted settings.

State evolution analysis confirms the algorithm's theoretical performance predictions.

Numerical experiments show UCS outperforms existing methods and characterizes detectability regions.

Abstract

This paper introduces an algorithmic solution to a broader class of unlabeled sensing problems with multiple measurement vectors (MMV). The goal is to recover an unknown structured signal matrix, $\mathbf{X}$, from its noisy linear observation matrix, $\mathbf{Y}$, whose rows are further randomly shuffled by an unknown permutation matrix $\mathbf{U}$. A new Bayes-optimal unlabeled compressed sensing (UCS) recovery algorithm is developed from the bilinear approximate message passing (Bi-VAMP) framework using non-separable and coupled priors on the rows and columns of the permutation matrix $\mathbf{U}$. In particular, standard unlabeled sensing is a special case of the proposed framework, and UCS further generalizes it by neither assuming a partially shuffled signal matrix $\mathbf{X}$ nor a small-sized permutation matrix $\mathbf{U}$. For the sake of theoretical performance prediction, we also conduct a state evolution (SE) analysis of the proposed algorithm and show its consistency with the asymptotic empirical mean-squared error (MSE). Numerical results demonstrate the effectiveness of the proposed UCS algorithm and its advantage over state-of-the-art baseline approaches in various applications. We also numerically examine the phase transition diagrams of UCS, thereby characterizing the detectability region as a function of the signal-to-noise ratio (SNR).