# Capacity lower bound for the Ising perceptron

@article{Ding2019CapacityLB, title={Capacity lower bound for the Ising perceptron}, author={Jian Ding and Nike Sun}, journal={Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing}, year={2019} }

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 MN for which the intersection is nonempty. It is conjectured by Krauth and Mézard (1989) that the (random) ratio MN/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 ≤ MN/N ≤ 1−γ with high probability; see also…

## 25 Citations

Proof of the Contiguity Conjecture and Lognormal Limit for the Symmetric Perceptron

- Mathematics2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

We consider the symmetric binary perceptron model, a simple model of neural networks that has gathered significant attention in the statistical physics, information theory and probability theory…

Critical Window of The Symmetric Perceptron

- Computer Science, Mathematics
- 2022

The critical window of the symmetric binary perceptron has nearly the “sharpest possible transition,” adding it to a short list of CSP for which the critical window is rigorously known to be of near-constant width.

Frozen 1-RSB structure of the symmetric Ising perceptron

- Computer ScienceSTOC
- 2021

It is proved, under an assumption on the critical points of a real-valued function, that the symmetric Ising perceptron exhibits the `frozen 1-RSB' structure; that is, typical solutions of the model lie in clusters of vanishing entropy density.

Algorithms and Barriers in the Symmetric Binary Perceptron Model

- Computer ScienceArXiv
- 2022

At high enough densities the symmetric binary perceptron exhibits the multi Overlap Gap Property ( m − OGP), an intricate geometrical property known to be a rigorous barrier for large classes of algorithms.

Storage capacity in symmetric binary perceptrons

- Computer ScienceJournal of Physics A: Mathematical and Theoretical
- 2019

The replica method is used to estimate the capacity threshold for the rectangle-binary-perceptron case when the u-function is wide and it is concluded that full-step-replica-symmetry breaking would have to be evaluated in order to obtain the exact capacity in this case.

Sharp threshold sequence and universality for Ising perceptron models

- Computer Science
- 2022

The results of this paper apply in more general settings, and are based on new “add one constraint” estimates extending Talagrand’s estimates for the half-space model (1999, 2011).

Sharp threshold for the Ising perceptron model

- Mathematics, Computer ScienceThe Annals of Probability
- 2021

It is proved that this event has a sharp threshold; that is, the probability that the intersection is empty increases quickly from $\epsilon$ to $1- \ep silon$ when $p$ increases only by a factor of $1 + o(1)$ as $N \to \infty$.

Binary perceptron: efficient algorithms can find solutions in a rare well-connected cluster

- Computer ScienceSTOC
- 2022

It is shown that at low constraint density, there exists indeed a subdominant connected cluster of solutions with almost maximal diameter, and that an efficient multiscale majority algorithm can find solutions in such a cluster with high probability, settling in particular an open problem posed by Perkins-Xu in STOC'21.

The discrepancy of random rectangular matrices

- MathematicsRandom Struct. Algorithms
- 2022

A complete answer to the Beck–Fiala conjecture is given for two natural models: matrices with Bernoulli or Poisson entries, and the discrepancy for any rectangular aspect ratio is characterized.

Algorithmic pure states for the negative spherical perceptron

- Computer ScienceArXiv
- 2020

An efficient algorithm is designed which, given oracle access to the solution of the Parisi variational principle, exploits this conjectured FRSB structure for $\kappa<0$ and outputs a vector $\sigma$ which is expected to be approximately the barycenter of a pure state of the spherical perceptron near criticality.

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