{"paper":{"title":"CLuP practically achieves $\\sim 1.77$ positive and $\\sim 0.33$ negative Hopfield model ground state free energy","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.IT","math.IT","math.OC","stat.ML"],"primary_cat":"cond-mat.dis-nn","authors_text":"Mihailo Stojnic","submitted_at":"2025-07-30T05:30:46Z","abstract_excerpt":"We study algorithmic aspects of finding $n$-dimensional \\emph{positive} and \\emph{negative} Hopfield ($\\pm$Hop) model ground state free energies. This corresponds to classical maximization of random positive/negative semi-definite quadratic forms over binary $\\left \\{\\pm \\frac{1}{\\sqrt{n}} \\right \\}^n$ vectors. The key algorithmic question is whether these problems can be computationally efficiently approximated within a factor $\\approx 1$. Following the introduction and success of \\emph{Controlled Loosening-up} (CLuP-SK) algorithms in finding near ground state energies of closely related Sher"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.22396","kind":"arxiv","version":1},"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/2507.22396/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"}