Constructive Cavity Method for Ising Restricted and Deep Boltzmann Machines with Gaussian Couplings
Abstract
We apply the Constructive Cavity Method (CCM) to the dense, Ising type, restricted and deep Boltzmann machine with Gaussian couplings between adjacent layers. After identifying the core and interface contributions, CCM identifies the blanket decomposition of the cavity variables with the Aizenman--Sims--Starr random overlap structure. We apply the Random Energy Model (REM) universality ansatz to represent the cavity free energies by those of a generalized random energy model (GREM) and evaluate them recursively through Poisson point processes (PPPs). In the layer-wise gauge each layer is cut with its own cumulative fraction |Uℓ∩ Lp|=λp qℓ(p)N+o(N), with 0=q0≤q1≤⋯≤qr=1, where qℓ=(qℓ(1),…,qℓ(K)). Here Uℓ is the cumulative blanket union up to blanket ℓ. The index p labels architectural layers, whereas ℓ labels the blankets. The dependence on the multilayer geometry is carried by the two matrices M=2M0Λ and M1=ΛM0Λ. The constructive cavity field in layer p has blanket variances Qℓ(p)−Qℓ−1(p)=(M(qℓ−qℓ−1))p, while the correction field has blanket variances Qℓ−Qℓ−1=qℓ⊤ M1qℓ−qℓ−1⊤ M1qℓ−1. The nested REM--PPP average gives the finite-level constructive cavity functional 𝒫RBM/DBMCCM(ζ,q)=log2+∑p=1KλpX0(p)−1/2∑ℓ=0r−1ζℓ(qℓ+1⊤ M1qℓ+1−qℓ⊤ M1qℓ). The balanced scalar gauge is recovered by imposing qℓ=qℓ1. Selecting the physical value in the non-convex multilayer regime is a separate problem.
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BibTeX
@misc{franchini2026constructive,
title = {{Constructive Cavity Method for Ising Restricted and Deep Boltzmann Machines with Gaussian Couplings}},
author = {Simone Franchini and Francesco Concetti and Giampiero Bardella and Liming Pan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36505},
eprint = {2610.36505},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36505},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Constructive Cavity Method for Ising Restricted and Deep Boltzmann Machines with Gaussian Couplings
AU - Simone Franchini
AU - Francesco Concetti
AU - Giampiero Bardella
AU - Liming Pan
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36505
UR - https://arxiv.org/abs/2610.36505
AB - We apply the Constructive Cavity Method (CCM) to the dense, Ising type, restricted and deep Boltzmann machine with Gaussian couplings between adjacent layers. After identifying the core and interface contributions, CCM identifies the blanket decomposition of the cavity variables with the Aizenman--Sims--Starr random overlap structure. We apply the Random Energy Model (REM) universality ansatz to represent the cavity free energies by those of a generalized random energy model (GREM) and evaluate them recursively through Poisson point processes (PPPs). In the layer-wise gauge each layer is cut with its own cumulative fraction |U_ℓ∩ L_p|=λ_p q_ℓ^(p)N+o(N), with 0=q_0≤q_1≤⋯≤q_r=1, where q_ℓ=(q_ℓ^(1),…,q_ℓ^(K)). Here U_ℓ is the cumulative blanket union up to blanket ℓ. The index p labels architectural layers, whereas ℓ labels the blankets. The dependence on the multilayer geometry is carried by the two matrices M=2M_0Λ and M_1=ΛM_0Λ. The constructive cavity field in layer p has blanket variances Q_ℓ^(p)−Q_ℓ−1^(p)=(M(q_ℓ−q_ℓ−1))_p, while the correction field has blanket variances Q_ℓ−Q_ℓ−1=q_ℓ^⊤ M_1q_ℓ−q_ℓ−1^⊤ M_1q_ℓ−1. The nested REM--PPP average gives the finite-level constructive cavity functional 𝒫_RBM/DBM^CCM(ζ,q)=log2+∑_p=1^Kλ_pX_0^(p)−1/2∑_ℓ=0^r−1ζ_ℓ(q_ℓ+1^⊤ M_1q_ℓ+1−q_ℓ^⊤ M_1q_ℓ). The balanced scalar gauge is recovered by imposing q_ℓ=q_ℓ1. Selecting the physical value in the non-convex multilayer regime is a separate problem.
ER -