TY - JOUR
T1 - On Convergence of the Cavity and Bolthausen’s TAP Iterations to the Local Magnetization
AU - Chen, Wei Kuo
AU - Tang, Si
N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2021/9
Y1 - 2021/9
N2 - 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.
AB - 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.
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U2 - 10.1007/s00220-021-04103-0
DO - 10.1007/s00220-021-04103-0
M3 - Article
AN - SCOPUS:85105862836
SN - 0010-3616
VL - 386
SP - 1209
EP - 1242
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
IS - 2
ER -