Capacity lower bound for the Ising perceptron

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).