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

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.