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

TL;DR

This paper rigorously solves a spin-glass model with mismatched inference involving Mattis interaction, providing a variational characterization and phase diagram for the reconstruction error.

Contribution

It introduces a rigorous solution to a mismatched spin-glass model with Mattis interaction using a variational approach, extending understanding of inference in complex systems.

Findings

Exact solution via variational principle for two order parameters

Identification of the Mattis magnetization concentration in the thermodynamic limit

Phase diagram for Gaussian signal distribution case

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 solution leads to the computation of the Mean Square Error of the mismatched reconstruction. The Gaussian signal distribution case is investigated and the corresponding phase diagram is identified.