Capacity lower bound for the Ising perceptron
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440_2025_Article_1364.pdf
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1.24 MB
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84f8f803daf6eae95935756485c53c41
Author(s)
Ding, Jian
Sun, Nike
Date Issued
February 23, 2025
Journal
Probability Theory and Related Fields
Publisher
Springer Berlin Heidelberg
Citation
Ding, J., Sun, N. Capacity lower bound for the Ising perceptron. Probab. Theory Relat. Fields 193, 627–715 (2025).
Version
Final published version
Abstract
We consider the Ising perceptron with gaussian disorder, which is equivalent to the discrete cube { - 1 , + 1 } N intersected by M random half-spaces. The perceptron’s capacity is the largest integer M N for which the intersection is nonempty. It is conjectured by Krauth and Mézard (1989) that the (random) ratio M N / N converges in probability to an explicit constant α ⋆ ≐ 0.83 . Kim and Roche (1998) proved the existence of a positive constant γ such that γ ⩽ M N / N ⩽ 1 - γ with high probability; see also Talagrand (1999). In this paper we show that the Krauth–Mézard conjecture α ⋆ is a lower bound with positive probability, under the condition that an explicit univariate function S ⋆ ( λ ) is maximized at λ = 0 . Our proof is an application of the second moment method to a certain slice of perceptron configurations, as selected by the so-called TAP (Thouless, Anderson, and Palmer, 1977) or AMP (approximate message passing) iteration, whose scaling limit has been characterized by Bayati and Montanari (2011) and Bolthausen (2012).
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1007/s00440-025-01364-x
