{"paper":{"title":"Belief Propagation and Loop Calculus for the Permanent of a Non-Negative Matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","cs.DM","cs.LG","cs.NA","math.OC"],"primary_cat":"cs.DS","authors_text":"Michael Chertkov, Yusuke Watanabe","submitted_at":"2009-11-08T04:15:01Z","abstract_excerpt":"We consider computation of permanent of a positive $(N\\times N)$ non-negative matrix, $P=(P_i^j|i,j=1,\\cdots,N)$, or equivalently the problem of weighted counting of the perfect matchings over the complete bipartite graph $K_{N,N}$. The problem is known to be of likely exponential complexity. Stated as the partition function $Z$ of a graphical model, the problem allows exact Loop Calculus representation [Chertkov, Chernyak '06] in terms of an interior minimum of the Bethe Free Energy functional over non-integer doubly stochastic matrix of marginal beliefs, $\\beta=(\\beta_i^j|i,j=1,\\cdots,N)$, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.1419","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":""},"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"}