The Multi-species Mean-Field Spin-Glass on the Nishimori Line

The Solution of the Deep Boltzmann Machine on the Nishimori Line

“…The proof of Theorem 1 relies on the adaptive interpolation method [8] combined with a concentration result and with the Nishimori identities, that will be presented in the next section. The main difference with the model solved in [2] is that the matrix Δ is not definite, indeed its eigenvalues have alternating signs. This entails that the remainder identified by interpolation has not a definite sign and cannot be discarded a priori at the expense of an inequality.…”

“…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. In the following, given a square matrix A we denote by ρ(A) its spectral radius and by A (eo) the submatrix of A obtained by keeping only even rows and odd columns of A.…”

“…The main thermodynamic properties of the model are consequences of a family of identities and inequalities for correlation functions that are due to the specific setting (2). The identities were introduced in the original work by Nishimori [25] while the inequalities were proved in [21,22].…”

“…This allows us to use the identities and inequalities for any value of the interpolating parameter. See Proof of Proposition 2 in [2] for the details.…”

“…While the general deep (restricted) Boltzmann machine is still an unsolved problem (see nevertheless [1,[3][4][5]12,18,23,24] for centered Gaussian interactions), we present here its exact and rigorous solution in a subregion of the phase space known as Nishimori line. In a previous paper [2] we have fully solved the elliptic multi-species model on the Nishimori line, where the property of replica symmetry, i.e. the concentration of the overlap, was shown to hold.…”