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

We analyse a linear regression problem with nonconvex regularization called smoothly clipped absolute deviation (SCAD) under an overcomplete Gaussian basis for Gaussian random data. We propose an approximate message passing (AMP) algorithm considering nonconvex regularization, namely SCAD-AMP, and analytically show that the stability condition corresponds to the de Almeida--Thouless condition in spin glass literature. Through asymptotic analysis, we show the correspondence between the density evolution of SCAD-AMP and the replica symmetric solution. Numerical experiments confirm that for a sufficiently large system size, SCAD-AMP achieves the optimal performance predicted by the replica method. Through replica analysis, a phase transition between replica symmetric (RS) and replica symmetry breaking (RSB) region is found in the parameter space of SCAD. The appearance of the RS region for a nonconvex penalty is a significant advantage that indicates the region of smooth landscape of the optimization problem. Furthermore, we analytically show that the statistical representation performance of the SCAD penalty is better than that of L1-based methods, and the minimum representation error under RS assumption is obtained at the edge of the RS/RSB phase. The correspondence between the convergence of the existing coordinate descent algorithm and RS/RSB transition is also indicated

The anchoring of inflation expectations in time and frequency domains

This paper introduces wavelet analysis as a tool for investigating the anchoring of inflation expectations in the United States. We show that the anchoring of inflation expectations varies for different groups of economic agents and changes across time and frequency. For consumers, we confirm significant lead-lag relationships between short- and long-term inflation expectations at medium frequencies of one to four years of scale, thus suggesting that short-term inflation expectations had fed into long-term inflation expectations over the crisis period. However, no such relationship is found for professional forecasters. These results indicate that long-term inflation expectations were de-anchored during the crisis period for consumers but not for professional forecasters. Although consumers’ long-term inflation expectations have been re-anchored since 2014 at medium frequencies, we find signs of de-anchoring at higher time scales of approximately eight year