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

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 $ρN$ ($0\le ρ\le 1$). Problems of this type can be solved by minimizing a cost function with respect to the $L_p$-norm $||\bx||_p=\lim_{ε\to +0}\sum_{i=1}^N |x_i|^{p+ε}$, subject to the constraints under an appropriate condition. For several $p$, we assess a typical case limit $α_c(ρ)$, which represents a critical relation between $α=P/N$ and $ρ$ for successfully reconstructing the original vector by minimization for typical situations in the limit $N,P \to \infty$ with keeping $α$ finite, utilizing the replica method. For $p=1$, $α_c(ρ)$ is considerably smaller than its worst case counterpart, which has been rigorously derived by existing literature of information theory.