{"paper":{"title":"Symmetric Perceptrons, Number Partitioning and Lattices","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CC","math-ph","math.MP","math.PR","stat.TH"],"primary_cat":"math.ST","authors_text":"Neekon Vafa, Vinod Vaikuntanathan","submitted_at":"2025-01-27T21:31:01Z","abstract_excerpt":"The symmetric binary perceptron ($\\mathrm{SBP}_{\\kappa}$) problem with parameter $\\kappa : \\mathbb{R}_{\\geq1} \\to [0,1]$ is an average-case search problem defined as follows: given a random Gaussian matrix $\\mathbf{A} \\sim \\mathcal{N}(0,1)^{n \\times m}$ as input where $m \\geq n$, output a vector $\\mathbf{x} \\in \\{-1,1\\}^m$ such that $$|| \\mathbf{A} \\mathbf{x} ||_{\\infty} \\leq \\kappa(m/n) \\cdot \\sqrt{m}~.$$ The number partitioning problem ($\\mathrm{NPP}_{\\kappa}$) corresponds to the special case of setting $n=1$. There is considerable evidence that both problems exhibit large computational-stat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.16517","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.16517/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}