An inference problem in a mismatched setting: a spin-glass model with Mattis interaction

Abstract

The Wigner spiked model in a mismatched setting is studied with the finite temperature Statistical Mechanics approach through its representation as a Sherrington-Kirkpatrick model with added Mattis interaction. The exact solution of the model with Ising spins is rigorously proved to be given by a variational principle on two order parameters, the Parisi overlap distribution and the Mattis magnetization. The latter is identified by an ordinary variational principle and turns out to concentrate in the thermodynamic limit. The Gaussian signal distribution case is investigated and the corresponding phase diagram is identified.

A Statistical Physics approach to a multi-channel Wigner spiked model

“…When instead the receiver ignores the prior or the signal-to-noise ratio, only a few results are available [20][21][22][23][24]. In such settings, called mismatched, the Nishimori identities do not hold true and a replica symmetry breaking variational principle may arise [25]. It is worth noticing that the Statistical Mechanics model described by the Hamiltonian (8) with centered interactions (which for K = 1 reduces to the standard Sherrington-Kirkpatrick model [26][27][28][29]) is still unsolved for non-convex architectures such as the deep one, see nevertheless [19,[30][31][32].…”