Critical Window of The Symmetric Perceptron
Dylan J. Altschuler

TL;DR
This paper proves that the symmetric binary perceptron exhibits a nearly sharp phase transition with a critical window of at most logarithmic size, significantly refining previous bounds and providing exponential tail bounds.
Contribution
It establishes that the critical window for the symmetric perceptron is at most O(log n), nearly the sharpest possible transition, with rigorous bounds and tail estimates.
Findings
Critical window is at most O(log n) in size.
CSP satisfiability sharply transitions at the critical density.
Provides exponential tail bounds for the transition.
Abstract
We study the critical window of the symmetric binary perceptron, or equivalently, combinatorial discrepancy. Consider the problem of finding a binary vector satisfying , where is an matrix with iid Gaussian entries. For fixed , at which densities is this constraint satisfaction problem (CSP) satisfiable? A sharp threshold was recently established by Perkins and Xu, and Abbe, Li, and Sly , answering this to first order. Namely, for each there exists an explicit critical density so that for any fixed , with high probability the CSP is satisfiable for and unsatisfiable for . This corresponds to a bound of on the size of the critical window. We sharpen these results significantly, as well as provide exponential…
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Taxonomy
TopicsMathematical Approximation and Integration · Digital Image Processing Techniques · semigroups and automata theory
