On Convergence of the Cavity and Bolthausen’s TAP Iterations to the Local Magnetization

Wei Kuo Chen, Si Tang

Research output: Contribution to journalArticlepeer-review

Abstract

The cavity and TAP equations are high-dimensional systems of nonlinear equations of the local magnetization in the Sherrington–Kirkpatrick model. In the seminal work, Bolthausen (Commun Math Phys 325(1):333–366, 2014) introduced an iterative scheme that produces an asymptotic solution to the TAP equations if the model lies inside the Almeida–Thouless transition line. However, it was unclear if this asymptotic solution coincides with the local magnetization. In this work, motivated by the cavity equations, we introduce a new iterative scheme and establish a weak law of large numbers. We show that our new scheme is asymptotically the same as the so-called approximate message passing algorithm, a generalization of Bolthausen’s iteration, that has been popularly adapted in compressed sensing, Bayesian inferences, etc. Based on this, we confirm that our cavity iteration and Bolthausen’s scheme both converge to the local magnetization as long as the overlap is locally uniformly concentrated.

Original languageEnglish (US)
Pages (from-to)1209-1242
Number of pages34
JournalCommunications in Mathematical Physics
Volume386
Issue number2
DOIs
StatePublished - Sep 2021

Bibliographical note

Funding Information:
Partly supported by NSF Grant DMS-17-52184

Funding Information:
Partly supported by the Collaboration Grant from the Simons Foundation #712728.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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