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We analyse a linear regression problem with nonconvex regularization calledsmoothly clipped absolute deviation (SCAD) under an overcomplete Gaussian basisfor Gaussian random data. We propose an approximate message passing (AMP)algorithm considering nonconvex regularization, namely SCAD-AMP, andanalytically show that the stability condition corresponds to the deAlmeida--Thouless condition in spin glass literature. Through asymptoticanalysis, we show the correspondence between the density evolution of SCAD-AMPand the replica symmetric solution. Numerical experiments confirm that for asufficiently large system size, SCAD-AMP achieves the optimal performancepredicted by the replica method. Through replica analysis, a phase transitionbetween replica symmetric (RS) and replica symmetry breaking (RSB) region isfound in the parameter space of SCAD. The appearance of the RS region for anonconvex penalty is a significant advantage that indicates the region ofsmooth landscape of the optimization problem. Furthermore, we analytically showthat the statistical representation performance of the SCAD penalty is betterthan that of L1-based methods, and the minimum representation error under RSassumption is obtained at the edge of the RS/RSB phase. The correspondencebetween the convergence of the existing coordinate descent algorithm and RS/RSBtransition is also indicated.

Approximate message passing for nonconvex sparse regularization with stability and asymptotic analysis