Existence and uniqueness of a TAP fixed point for the Ising perceptron

Establish existence and uniqueness, in the true (unconditioned) Ising perceptron model, of a stationary point (m,n) of the Thouless–Anderson–Palmer (TAP) free energy associated with the model, thereby justifying a rigorous link between the true model and the planted TAP formulation.

Background

The paper’s reduction to a planted model hinges on conditioning on a TAP fixed point (m,n). While this makes the analysis tractable, in the true model the TAP solution depends on the disorder matrix in a complicated way.

A rigorous equivalence between the planted construction and the true model would follow from existence and uniqueness of the TAP fixed point, which the authors note is not currently established for the Ising perceptron.