Unlabeled Compressed Sensing from Multiple Measurement Vectors

TL;DR

This work tackles unlabeled compressed sensing for MMV by formulating it as a bilinear recovery problem with an unknown permutation matrix $\mathbf{U}$. It introduces UCS, an AMP-based algorithm that jointly recovers $\mathbf{X}$ and $\mathbf{U}$ by deploying two coupled denoisers for the rows and columns of $\mathbf{U}$ within the Bi-VAMP framework, thereby accommodating non-separable priors on $\mathbf{U}$. A thorough state evolution analysis confirms that UCS predicts the asymptotic MSE and LLR behavior, with simulations showing strong performance gains over baselines and clear phase transition behavior as SNR, dimensions, and sparsity vary. The method unifies unlabeled sensing and unlabeled CS under a Bayes-optimal inference paradigm, delivering practical algorithms for large-scale problems and providing theoretical performance guarantees. The study demonstrates UCS’s potential across applications requiring permutation recovery and robust signal reconstruction from shuffled measurements.

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).

Figures (8)

• Figure 1: Factor graph of Bi-VAMP and its outgoing messages that are used by the denoisers of $\bm{u}_i$ and $\bm{v}_j$ (taken from akrout2020bilinearArxiv).
• Figure 2: Block diagram of the proposed UCS recovery algorithm with its three modules: the two prior modules $p_{\textrm{{U}}^+}(.)$ and $p_{\textrm{{X}}^+}(.)$, and the Bi-LMMSE module. The latter exchanges extrinsic information/messages with the prior modules $p_{\textrm{{U}}^+}(.)$ and $p_{\textrm{{X}}^+}(.)$ through the and blocks, respectively. The LLR messages (to the left of Bi-LMMSE) are calculated using belief propagation while denoising $\boldsymbol{U}^+$. The Gaussian messages (to the right of Bi-LMMSE) are calculated using the expectation propagation principle while denoising $\boldsymbol{X}^+$. The color of each module matches the color of the corresponding line numbers in Algorithm \ref{['algo:big-vamp']}.
• Figure 3: State evolution diagram of (a) Bi-VAMP where only an assignment prior on $\bm{U}^-$ is enforced, and (b) UCS where a row-wise prior on $\bm{U}^-$ and a column-wise prior on $\bm{U}^+$ are enforced. The color of each MSE function in (b) matches the color of the corresponding module in the block diagram in Fig. \ref{['fig:block-diagram']} in which it is involved.
• Figure 4: HD of UCS vs. the size $p$ of the diagonal blocks of $\bm{U}$ for the $p$-local unlabeled sensing problem at SNR = 20 dB: Gaussian prior on $\bm{X}$ with $N=160$, $M=100$, and $R=10$.
• Figure 5: UCS reconstruction performance of the matrices $\bm{U}$ and $\bm{X}$ for the unlabeled sensing problem over a grid of $N$ and $M$ values between 60 and 150 at SNR = 30 dB: Gaussian prior on $\bm{X}$ and $R=10$.