Statistical Mechanical Analysis of a Typical Reconstruction Limit of Compressed Sensing

TL;DR

This paper employs the replica method from statistical mechanics to analyze the typical reconstruction limits of compressed sensing, revealing a universal critical relation between measurement ratio and sparsity under certain matrix conditions.

Contribution

It introduces a statistical mechanical framework to derive a universal reconstruction limit in compressed sensing, extending previous empirical observations.

Findings

Derives a universal critical relation between measurement ratio and sparsity.

Shows the relation holds for matrices characterized by rotationally invariant ensembles.

Supports the universality of reconstruction limits across various matrix ensembles.

Abstract

We use the replica method of statistical mechanics to examine a typical performance of correctly reconstructing $N$-dimensional sparse vector $bx=(x_i)$ from its linear transformation $by=bF bx$ of $P$ dimensions on the basis of minimization of the $L_p$-norm $||bx||_p= lim_{epsilon to +0} sum_{i=1}^N |x_i|^{p+epsilon}$. We characterize the reconstruction performance by the critical relation of the successful reconstruction between the ratio $alpha=P/N$ and the density $rho$ of non-zero elements in $bx$ in the limit $P,,N to infty$ while keeping $alpha sim O(1)$ and allowing asymptotically negligible reconstruction errors. We show that the critical relation $alpha_c(rho)$ holds universally as long as $bF^{rm T}bF$ can be characterized asymptotically by a rotationally invariant random matrix ensemble and $bF bF^{rm T}$ is typically of full rank. This supports the universality of the critical relation observed by Donoho and Tanner ({em Phil. Trans. R. Soc. A}, vol.~367, pp.~4273--4293, 2009; arXiv: 0807.3590) for various ensembles of compression matrices.