  We consider compressed sensing formulated as a minimization problem of
nonconvex sparse penalties, Smoothly Clipped Absolute deviation (SCAD) and
Minimax Concave Penalty (MCP). The nonconvexity of these penalties is
controlled by nonconvexity parameters, and L1 penalty is contained as a limit
with respect to these parameters. The analytically derived reconstruction limit
overcomes that of L1 and the algorithmic limit in the Bayes-optimal setting,
when the nonconvexity parameters have suitable values. However, for small
nonconvexity parameters, where the reconstruction of the relatively dense
signals is theoretically guaranteed, the corresponding approximate message
passing (AMP) cannot achieve perfect reconstruction. We identify that the
shrinks in the basin of attraction to the perfect reconstruction causes the
discrepancy between the AMP and corresponding theory using state evolution. A
part of the discrepancy is resolved by introducing the control of the
nonconvexity parameters to guide the AMP trajectory to the basin of the
attraction.
