Replica Symmetry Breaking in Multi-species Sherrington–Kirkpatrick Model

Abstract

In the Sherrington-Kirkpatrick (SK) and related mixed p-spin models, there is interest in understanding replica symmetry breaking at low temperatures. For this reason, the so-called AT line proposed by de Almeida and Thouless as a sufficient (and conjecturally necessary) condition for symmetry breaking, has been a frequent object of study in spin glass theory. In this paper, we consider the analogous condition for the multi-species SK model, which concerns the eigenvectors of a Hessian matrix. The analysis is tractable in the two-species case with positive definite variance structure, for which we derive an explicit AT temperature threshold. To our knowledge, this is the first nonasymptotic symmetry breaking condition produced for a multi-species spin glass. As possible evidence that the condition is sharp, we draw further parallel with the classical SK model and show coincidence with a separate temperature inequality guaranteeing uniqueness of the replica symmetric critical point.2010 Mathematics Subject Classification. 60K35, 82B26, 82B44.

The Solution of the Deep Boltzmann Machine on the Nishimori Line

“…The previous implies that the matrix [− 1 + S] (oo) in (93) is negative definite, making H x o π negative definite too, and hence π is concave under the hypothesis ρ([M 2 ] (oo) ) < 1. In turn, this ensures uniqueness of the solution to the consistency equation (84) and to the variational problem (13). In particular when h = 0, x = 0 is the unique solution.…”

“…Secondly, we focus on the properties of the consistency equation obtained from the optimization problem (13) when the matrix Δ is invertible, that is when K is even. The stability of the optimizers of ( 13) is a more delicate problem with respect to the convex multi-species case [2], due to the min-max nature of the variational principle.…”

“…Let K be even and h = 0. If ρ([M 2 ] (oo) ) < 1 then x = 0 is the unique solution to the variational problem (13). Conversely, if ρ([M 2 ] (oo) ) > 1 then the solution of (13) is a vector x =x(M) with strictly positive components satisfying the consistency equation:…”

“…Now we move to the proof of the variational principle, i.e. the second equality in (13) which is going to be achieved through upper and lower bounds. For what follows, we neglect all the sub and superscripts in the Boltzmann-Gibbs averages, except for the t-dependence.…”

“…A deep (restricted) Boltzmann machine can be considered as a special case of the mean field multi-species spin glass model introduced in [11], further studied in [13,27]. Specifically the set of spins is arranged into a geometry made of consecutive layers and only interactions among spins belonging to adjacent layers are allowed.…”