Sharp threshold sequence and universality for Ising perceptron models

Sharp threshold for the Ising perceptron model

Consider the discrete cube $\{-1,1\}^N$ and a random collection of half spaces which includes each half space $H(x) := \{y \in \{-1,1\}^N: x \cdot y \geq κ\sqrt{N}\}$ for $x \in \{-1,1\}^N$ independently with probability $p$. Is the intersection of these half spaces empty? This is called the Ising perceptron model under Bernoulli disorder. We prove that this event has a sharp threshold; that is, the probability that the intersection is empty increases quickly from $ε$ to $1- ε$ when $p$ increases only by a factor of $1 + o(1)$ as $N \to \infty$.

Ising感知机的副本对称鞍点唯一性

Uniqueness of Replica-symmetric Saddle Point for Ising Perceptron

本文证明了在特定边缘条件下，Ising感知机的副本对称鞍点方程在临界容量以下有唯一解，并提供了完全解析的证明方法。

Frozen $1$-RSB structure of the symmetric Ising perceptron

We prove, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the `frozen 1-RSB' structure conjectured by Krauth and Mezard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1-RSB scenario is part of a recent and intriguing explanat…

Gardner formula for Ising perceptron models at small densities

We consider the Ising perceptron model with N spins and M = N*alpha patterns, with a general activation function U that is bounded above. For U bounded away from zero, or U a one-sided threshold function, it was shown by Talagrand (2000, 2011) that for small densities alpha, the free energy of the model converges in the large-N limit to the replica symmetric formula conjectured in the physics literature (Krauth--Mezard 1989, see also Gardner--Derrida 1988). We give a new proof of this result, which covers the more general class of all functions U that are bounded above and satisfy a certain va…

伊辛感知器的存储容量：对卓越数值条件的验证

The storage capacity of the Ising perceptron: verification of the outstanding numerical conditions

研究伊辛感知器存储容量，通过计算机辅助证明MN/N依概率收敛于α★。利用Arb球算术严格验证丁 - 孙和黄的条件，重新建立共享参数矩形并验证丁 - 孙条件，结合相关定理证明高斯及多种无序定律下的结果。

Ising感知机的容量阈值

Capacity threshold for the Ising perceptron

本文在特定数值条件下证明了Ising感知机容量以高概率不超过Krauth-Mézard猜想常数0.833，与已有下界结果共同给出该猜想的条件性证明。