# 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}
}
• Published 20 September 2018
• Mathematics, Computer Science
• Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing
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…

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## References

SHOWING 1-10 OF 45 REFERENCES
Covering Cubes by Random Half Cubes with Applications to Binary Neural Networks
• Computer Science
J. Comput. Syst. Sci.
• 1998
LetQnbe the (hyper)cube {?1, 1}n. This paper is concerned with the following question: How many vectors must be chosen uniformly and independently at random fromQnbefore every vector inQnitself has
Universality of the SAT-UNSAT (jamming) threshold in non-convex continuous constraint satisfaction problems
• Computer Science
• 2017
It is conjecture that there is a large universality class of non-convex continuous CSPs whose SAT-UNSAT threshold is described by the same scaling solution, and it is proposed that the perceptron is the simplest prototype of these problems.
TAP free energy, spin glasses and variational inference
• Physics
The Annals of Probability
• 2021
We consider the Sherrington-Kirkpatrick model of spin glasses with ferromagnetically biased couplings. For a specific choice of the couplings mean, the resulting Gibbs measure is equivalent to the
The space of interactions in neural network models
The typical fraction of the space of interactions between each pair of N Ising spins which solve the problem of storing a given set of p random patterns as N-bit spin configurations is considered.
High dimensional robust M-estimation: asymptotic variance via approximate message passing
• Computer Science
ArXiv
• 2013
It is shown here that that this phenomenon can be characterized rigorously using techniques that were developed by the authors for analyzing the Lasso estimator under high-dimensional asymptotics, and clarified that the ‘extra Gaussian noise’ encountered in this problem is fundamentally similar to phenomena already studied for regularized least squares in the setting of n.
Finite-sample analysis of Approximate Message Passing
• Computer Science
2016 IEEE International Symposium on Information Theory (ISIT)
• 2016
This paper derives a concentration result for AMP with i.i.d. Gaussian measurement matrices with finite dimension n × N, and shows that the probability of deviation from the state evolution prediction falls exponentially in n.
Local entropy as a measure for sampling solutions in Constraint Satisfaction Problems
• Computer Science
• 2015
A novel entropy-driven Monte Carlo strategy to efficiently sample solutions of random constraint satisfaction problems (CSPs) and a fast solver that relies exclusively on a local entropy estimate is constructed, and can be applied to general CSPs.
The simplest model of jamming
• Computer Science
• 2015
It is shown that isostaticity is not a sufficient condition for singular force and gap distributions, and universality is hypothesized for a large class of non-convex constrained satisfaction problems with continuous variables.
The Dynamics of Message Passing on Dense Graphs, with Applications to Compressed Sensing
• Computer Science
IEEE Transactions on Information Theory
• 2010
This paper proves that indeed it holds asymptotically in the large system limit for sensing matrices with independent and identically distributed Gaussian entries, and provides rigorous foundation to state evolution.
The Thermodynamic Limit in Mean Field Spin Glass Models
• Mathematics
• 2002
Abstract: We present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities, for a large class of mean field disordered models, as