Abstract
We study the critical window of the symmetric binary perceptron, or equivalently, random combinatorial discrepancy. Consider the problem of finding a -valued vector σ satisfying , where A is an matrix with iid Gaussian entries. For fixed K, at which constraint densities α is this constraint satisfaction problem (CSP) satisfiable? A sharp threshold was recently established by Perkins and Xu [29], and Abbe, Li, and Sly [2], answering this to first order. Namely, for each K 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 tail bounds. Our main result is that, perhaps surprisingly, the critical window is actually at most of order . More precisely, for a large constant C, with high probability the CSP is satisfiable for and unsatisfiable for . These results add the the symmetric perceptron to the short list of CSP models for which a critical window is rigorously known, and to the even shorter list for which this window is known to have nearly constant width.
Acknowledgments
We thank Jonathan Niles-Weed for invaluable conversations and feedback at all stages of this manuscript. We thank Will Perkins for insightful feedback on a previous draft. We are also grateful to the anonymous reviewers. This work was funded in part by an NYU MacCracken fellowship, NSF Graduate Research Fellowship Program grant DGE-1839302, and NSF grant DMS-2015291.
Citation
Dylan J. Altschuler. "Critical window of the symmetric perceptron." Electron. J. Probab. 28 1 - 28, 2023. https://doi.org/10.1214/23-EJP1024
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