{"paper":{"title":"Two equalities expressing the determinant of a matrix in terms of expectations over matrix-vector products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"stat.CO","authors_text":"Jascha Sohl-Dickstein","submitted_at":"2020-05-13T19:42:40Z","abstract_excerpt":"We introduce two equations expressing the inverse determinant of a full rank matrix $\\mathbf{A} \\in \\mathbb{R}^{n \\times n}$ in terms of expectations over matrix-vector products. The first relationship is $|\\mathrm{det} (\\mathbf{A})|^{-1} = \\mathbb{E}_{\\mathbf{s} \\sim \\mathcal{S}^{n-1}}\\bigl[\\, \\Vert \\mathbf{As}\\Vert^{-n} \\bigr]$, where expectations are over vectors drawn uniformly on the surface of an $n$-dimensional radius one hypersphere. The second relationship is $|\\mathrm{det}(\\mathbf{A})|^{-1} = \\mathbb{E}_{\\mathbf{x} \\sim q}[\\,p(\\mathbf{Ax}) /\\, q(\\mathbf{x})]$, where $p$ and $q$ are s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.06553","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/2005.06553/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"}