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 $$ U(x)=\int_{\mathbb{R}} e^{tx}ρ(\mathrm{d}t),\qquad ρ\ge0, $$ 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, $Q_\ell$ on the Ising side and $Γ_\ell$ on the auxiliary side. The bipartite shell scale is $$ Q_{\ell+1}Γ_{\ell+1}-Q_\ellΓ_\ell, $$ and Gaussian core comparison removes the core contribution $ΔQ_\ellΔΓ_\ell$ under blanket refinement. The scalar cavity functional has three terms, $$ \mathcal{A}_{\mathrm{perc}}(ν)=\mathbb{E}\left[\log\langle2\cosh x\rangle_ν+α\log\langle U(s)\rangle_ν-\log\langle e^y\rangle_ν\right], $$ 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 $$ \mathcal{G}_{\mathrm{perc}}^{\mathrm{CCM}}(ζ,Q,Γ)=X_0(0)+αY_0(0)-\frac{1}{2}\sum_{\ell=0}^{R}ζ_\ell\left(Q_{\ell+1}Γ_{\ell+1}-Q_\ellΓ_\ell\right). $$ 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.