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In many application areas we are faced with the following question: Can werecover a sparse vector $x_o \in \mathbb{R}^N$ from its undersampled set ofnoisy observations $y \in \mathbb{R}^n$, $y=A x_o+w$. The last decade haswitnessed a surge of algorithms and theoretical results addressing thisquestion. One of the most popular algorithms is the $\ell_p$-regularized leastsquares (LPLS) given by the following formulation: \[ \hat{x}(\gamma,p )\in\arg\min_x \frac{1}{2}\|y - Ax\|_2^2+\gamma\|x\|_p^p, \] where $p \in [0,1]$.Despite the non-convexity of these problems for $p<1$, they are still appealingbecause of the following folklores in compressed sensing: (i) $\hat{x}(\gamma,p)$ is closer to $x_o$ than $\hat{x}(\gamma,1)$. (ii) If we employ iterativemethods that aim to converge to a local minima of LPLS, then under goodinitialization these algorithms converge to a solution that is closer to $x_o$than $\hat{x}(\gamma,1)$. In spite of the existence of plenty of empiricalresults that support these folklore theorems, the theoretical progress toestablish them has been very limited.  This paper aims to study the above folklore theorems and establish theirscope of validity. Starting with approximate message passing algorithm as aheuristic method for solving LPLS, we study the impact of initialization on theperformance of AMP. Then, we employ the replica analysis to show the connectionbetween the solution of AMP and $\hat{x}(\gamma, p)$ in the asymptoticsettings. This enables us to compare the accuracy of $\hat{x}(\gamma,p)$ for $p\in [0,1]$. In particular, we will characterize the phase transition and noisesensitivity of LPLS for every $0\leq p\leq 1$ accurately. Our results in thenoiseless setting confirm that LPLS exhibits the same phase transition forevery $0\leq p <1$ and this phase transition is much higher than that of LASSO.

Does $\ell_p$-minimization outperform $\ell_1$-minimization?