A typical reconstruction limit of compressed sensing based on Lp-norm minimization

TL;DR

This paper analyzes the fundamental limits of reconstructing sparse signals using Lp-norm minimization in compressed sensing, revealing typical case thresholds that outperform worst-case bounds for successful recovery.

Contribution

It provides a theoretical assessment of the typical reconstruction limit in compressed sensing using Lp-norm minimization, employing the replica method for various p values.

Findings

For p=1, the typical case threshold is significantly lower than the worst-case bound.

The study characterizes the critical measurement ratio () for successful reconstruction.

Results are derived in the limit of large N and P, with finite .

Abstract

We consider the problem of reconstructing an $N$-dimensional continuous vector $\bx$ from $P$ constraints which are generated by its linear transformation under the assumption that the number of non-zero elements of $\bx$ is typically limited to $\rho N$ ($0\le \rho \le 1$). Problems of this type can be solved by minimizing a cost function with respect to the $L_p$-norm $||\bx||_p=\lim_{\epsilon \to +0}\sum_{i=1}^N |x_i|^{p+\epsilon}$, subject to the constraints under an appropriate condition. For several $p$, we assess a typical case limit $\alpha_c(\rho)$, which represents a critical relation between $\alpha=P/N$ and $\rho$ for successfully reconstructing the original vector by minimization for typical situations in the limit $N,P \to \infty$ with keeping $\alpha$ finite, utilizing the replica method. For $p=1$, $\alpha_c(\rho)$ is considerably smaller than its worst case counterpart, which has been rigorously derived by existing literature of information theory.