A statistical physics approach to a multi-channel Wigner spiked model

In this letter we present a finite temperature approach to a high-dimensional inference problem, the Wigner spiked model, with group dependent signal-to-noise ratios. For two classes of convex and non-convex network architectures the error in the reconstruction is described in terms of the solution of a mean-field spin-glass on the Nishimori line. In the cases studied the order parameters do not fluctuate and are the solution of finite dimensional variational problems. The deep architecture is optimized in order to confine the high temperature phase where reconstruction fails.