The Constructive Cavity Method for the Ising Perceptron
Abstract
We apply the Constructive Cavity Method (CCM) to the Ising perceptron. The process starts from positive soft channels admitting an auxiliary representation U(x)=∫ℝ etxρ(dt), ρ≥0, with an auxiliary measure having a finite exponential quadratic moment. This rewrites the perceptron as a dense, bipartite Gaussian model with an Ising layer and a continuous auxiliary layer. The CCM blanket geometry then gives two constructive variance profiles, Qℓ on the Ising side and Γℓ on the auxiliary side. The bipartite shell scale is Qℓ+1Γℓ+1−QℓΓℓ, and Gaussian core comparison removes the core contribution ΔQℓΔΓℓ under blanket refinement. The scalar cavity functional has three terms, 𝒜perc(ν)=𝔼[log⟨2cosh x⟩ν+αlog⟨ U(s)⟩ν−log⟨ ey⟩ν], where x is the new-spin field, s is the new-constraint field, and y is the cavity correction. CCM identifies the constructive cavity variables with the Aizenman--Sims--Starr random overlap structure (ROSt) variables. Parisi's recursive averaging then gives 𝒢percCCM(ζ,Q,Γ)=X0(0)+αY0(0)−1/2∑ℓ=0Rζℓ(Qℓ+1Γℓ+1−QℓΓℓ). The replica-symmetric specialization gives Gardner's formula for the generalized Ising perceptron. Specializing the constraint terminal to the hard channel gives the Krauth--Mézard functional for the binary perceptron.
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BibTeX
@misc{franchini2026constructive,
title = {{The Constructive Cavity Method for the Ising Perceptron}},
author = {Simone Franchini and Francesco Concetti and Giampiero Bardella},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39141},
eprint = {2610.39141},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39141},
}
RIS
TY - UNPB
TI - The Constructive Cavity Method for the Ising Perceptron
AU - Simone Franchini
AU - Francesco Concetti
AU - Giampiero Bardella
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39141
UR - https://arxiv.org/abs/2610.39141
AB - We apply the Constructive Cavity Method (CCM) to the Ising perceptron. The process starts from positive soft channels admitting an auxiliary representation U(x)=∫_ℝ e^txρ(dt), ρ≥0, with an auxiliary measure having a finite exponential quadratic moment. This rewrites the perceptron as a dense, bipartite Gaussian model with an Ising layer and a continuous auxiliary layer. The CCM blanket geometry then gives two constructive variance profiles, Q_ℓ on the Ising side and Γ_ℓ on the auxiliary side. The bipartite shell scale is Q_ℓ+1Γ_ℓ+1−Q_ℓΓ_ℓ, and Gaussian core comparison removes the core contribution ΔQ_ℓΔΓ_ℓ under blanket refinement. The scalar cavity functional has three terms, 𝒜_perc(ν)=𝔼[log⟨2cosh x⟩_ν+αlog⟨ U(s)⟩_ν−log⟨ e^y⟩_ν], where x is the new-spin field, s is the new-constraint field, and y is the cavity correction. CCM identifies the constructive cavity variables with the Aizenman--Sims--Starr random overlap structure (ROSt) variables. Parisi's recursive averaging then gives 𝒢_perc^CCM(ζ,Q,Γ)=X_0(0)+αY_0(0)−1/2∑_ℓ=0^Rζ_ℓ(Q_ℓ+1Γ_ℓ+1−Q_ℓΓ_ℓ). The replica-symmetric specialization gives Gardner's formula for the generalized Ising perceptron. Specializing the constraint terminal to the hard channel gives the Krauth--Mézard functional for the binary perceptron.
ER -