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Capacity threshold for the Ising perceptron

Brice Huang

TL;DR

The paper establishes a conditional upper bound on the Ising perceptron capacity by connecting the capacity problem to a TAP/AMP mean-field framework and a planted-model reduction. Central to the approach is the AMP-conditioned moment method, which is extended via approximate contiguity to a planted TAP solution, enabling a tractable two-parameter first-moment optimization governed by the function 𝒮⋆(λ1,λ2). Under numerical conditions including 𝒮⋆(λ1,λ2) ≤ 0 with equality at (1,0) and a robust local concavity around late AMP iterates, the authors prove that for α>α⋆(κ) the probability M_N(κ)/N≥α tends to 0, and for κ=0 the result aligns with Krauth–Mézard’s conjectured threshold. The analysis combines TAP/AMP formulas, state evolution, determinant concentration, and a detailed planted-model first-moment evaluation, with a computer-assisted verification of the key numerical conditions. Together, these results deliver a conditional, near-rigorous endorsement of the Krauth–Mézard capacity threshold for the κ=0 Ising perceptron and illustrate a versatile methodology for sharp thresholds in non-convex mean-field problems.

Abstract

We show that the capacity of the Ising perceptron is with high probability upper bounded by the constant $α_\star \approx 0.833$ conjectured by Krauth and Mézard, under the condition that an explicit two-variable function $\mathscr{S}_*(λ_1,λ_2)$ is maximized at $(1,0)$. The earlier work of Ding and Sun proves the matching lower bound subject to a similar numerical condition, and together these results give a conditional proof of the conjecture of Krauth and Mézard.

Capacity threshold for the Ising perceptron

TL;DR

The paper establishes a conditional upper bound on the Ising perceptron capacity by connecting the capacity problem to a TAP/AMP mean-field framework and a planted-model reduction. Central to the approach is the AMP-conditioned moment method, which is extended via approximate contiguity to a planted TAP solution, enabling a tractable two-parameter first-moment optimization governed by the function 𝒮⋆(λ1,λ2). Under numerical conditions including 𝒮⋆(λ1,λ2) ≤ 0 with equality at (1,0) and a robust local concavity around late AMP iterates, the authors prove that for α>α⋆(κ) the probability M_N(κ)/N≥α tends to 0, and for κ=0 the result aligns with Krauth–Mézard’s conjectured threshold. The analysis combines TAP/AMP formulas, state evolution, determinant concentration, and a detailed planted-model first-moment evaluation, with a computer-assisted verification of the key numerical conditions. Together, these results deliver a conditional, near-rigorous endorsement of the Krauth–Mézard capacity threshold for the κ=0 Ising perceptron and illustrate a versatile methodology for sharp thresholds in non-convex mean-field problems.

Abstract

We show that the capacity of the Ising perceptron is with high probability upper bounded by the constant conjectured by Krauth and Mézard, under the condition that an explicit two-variable function is maximized at . The earlier work of Ding and Sun proves the matching lower bound subject to a similar numerical condition, and together these results give a conditional proof of the conjecture of Krauth and Mézard.
Paper Structure (50 sections, 71 theorems, 229 equations, 1 figure)

This paper contains 50 sections, 71 theorems, 229 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Notation
  4. Further background and proof outline
  5. AMP-conditioned moment method
  6. Approximate contiguity with planted model
  7. Topological trivialization of TAP free energy
  8. Critical point near late AMP iterates and determinant concentration
  9. On planted models
  10. First moment in planted model
  11. Formal statement of results
  12. Krauth--Mézard threshold
  13. Main result
  14. Proof of Theorem \ref{['thm:main']}
  15. TAP and AMP formulas
  16. ...and 35 more sections

Key Result

Theorem 1.1

ding2018capacity Under Condition 1.2 therein, the following holds for the $\kappa = 0$ Ising perceptron. For any $\alpha < \alpha_\star$, $\liminf_{N\to\infty} \mathop{\mathrm{{\mathbb{P}}}}\limits(M_N/N \ge \alpha) > 0$.

Figures (1)

  • Figure 1: Plots of $(x,y) \mapsto {\overline {\mathscr{S}}}_\star({\mathrm{th}}^{-1}(x),{\mathrm{th}}^{-1}(y))$ for $\kappa = 0$. Figure \ref{['subfig:plot-main']} plots over $x,y \in [-1,1]^2$, while Figure \ref{['subfig:plot-zoom']} restricts to inputs with ${\overline {\mathscr{S}}}_\star({\mathrm{th}}^{-1}(x),{\mathrm{th}}^{-1}(y)) \ge -0.01$. The plots lie below $0$, and from Figure \ref{['subfig:plot-zoom']} it appears the unique maximizer is $(x,y) = ({\mathrm{th}}(1),0)$, corresponding to $(\lambda_1,\lambda_2) = (1,0)$.

Theorems & Definitions (178)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5: Proved in §\ref{['sec:planted-mt']}
  • Claim 2.6: Proved in Appendix \ref{['app:numerics']}
  • Proposition 3.2: ding2018capacity
  • Lemma 3.5: Proved in §\ref{['sec:local-concavity']}
  • ...and 168 more