{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:QTE2A5YIWVB4FQLXHVD6ENT73Y","short_pith_number":"pith:QTE2A5YI","canonical_record":{"source":{"id":"2005.06553","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2020-05-13T19:42:40Z","cross_cats_sorted":[],"title_canon_sha256":"3b0c2af856ba2e94297c17443636091a3ccd1e6e469c6a128a3595bfa7b861ee","abstract_canon_sha256":"59bdd752704377f7390321a3f7521ca746dd08ca6308a3e5d3c06a3dd638dc58"},"schema_version":"1.0"},"canonical_sha256":"84c9a07708b543c2c1773d47e2367fde3fc46a5921303a271fa350c117f893a0","source":{"kind":"arxiv","id":"2005.06553","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.06553","created_at":"2026-07-05T01:11:34Z"},{"alias_kind":"arxiv_version","alias_value":"2005.06553v2","created_at":"2026-07-05T01:11:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.06553","created_at":"2026-07-05T01:11:34Z"},{"alias_kind":"pith_short_12","alias_value":"QTE2A5YIWVB4","created_at":"2026-07-05T01:11:34Z"},{"alias_kind":"pith_short_16","alias_value":"QTE2A5YIWVB4FQLX","created_at":"2026-07-05T01:11:34Z"},{"alias_kind":"pith_short_8","alias_value":"QTE2A5YI","created_at":"2026-07-05T01:11:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:QTE2A5YIWVB4FQLXHVD6ENT73Y","target":"record","payload":{"canonical_record":{"source":{"id":"2005.06553","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2020-05-13T19:42:40Z","cross_cats_sorted":[],"title_canon_sha256":"3b0c2af856ba2e94297c17443636091a3ccd1e6e469c6a128a3595bfa7b861ee","abstract_canon_sha256":"59bdd752704377f7390321a3f7521ca746dd08ca6308a3e5d3c06a3dd638dc58"},"schema_version":"1.0"},"canonical_sha256":"84c9a07708b543c2c1773d47e2367fde3fc46a5921303a271fa350c117f893a0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:11:34.328488Z","signature_b64":"oQKQlOX+b7oPKgf54FpWUuh/amLkAyMQw2k2XuB+UakuEOfWPxkYzTu9ZGqcCnn6he+dBlNSIQ7XJ+bWCjbtAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"84c9a07708b543c2c1773d47e2367fde3fc46a5921303a271fa350c117f893a0","last_reissued_at":"2026-07-05T01:11:34.328086Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:11:34.328086Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2005.06553","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:11:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TcBo67HAiOe1ZbCEwWhbqzha9r3gYBFP9vbHMtyKPXb/PbRp3/i7A55nuPSKlBjJ4gGzHjW8JXas6GOD6OQ7DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-20T11:54:35.821483Z"},"content_sha256":"14b88d781b6e080ff1ff04b6113558499123b4d3a2278f6654d3b14b7834a404","schema_version":"1.0","event_id":"sha256:14b88d781b6e080ff1ff04b6113558499123b4d3a2278f6654d3b14b7834a404"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:QTE2A5YIWVB4FQLXHVD6ENT73Y","target":"graph","payload":{"graph_snapshot":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:11:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8lrtTOfNXl52+8DoPDbn5SVpd6oWjbhhhFII5TOz9eB4j/0NpE2lV7pZGOG4gHu0ZOmakZ3YgVcER5n1FE63DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-20T11:54:35.821865Z"},"content_sha256":"5b857e1d9fffb9fc708e71124358b49518d51ac50f01e2bd49647ed483ab1443","schema_version":"1.0","event_id":"sha256:5b857e1d9fffb9fc708e71124358b49518d51ac50f01e2bd49647ed483ab1443"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QTE2A5YIWVB4FQLXHVD6ENT73Y/bundle.json","state_url":"https://pith.science/pith/QTE2A5YIWVB4FQLXHVD6ENT73Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QTE2A5YIWVB4FQLXHVD6ENT73Y/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-20T11:54:35Z","links":{"resolver":"https://pith.science/pith/QTE2A5YIWVB4FQLXHVD6ENT73Y","bundle":"https://pith.science/pith/QTE2A5YIWVB4FQLXHVD6ENT73Y/bundle.json","state":"https://pith.science/pith/QTE2A5YIWVB4FQLXHVD6ENT73Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QTE2A5YIWVB4FQLXHVD6ENT73Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:QTE2A5YIWVB4FQLXHVD6ENT73Y","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"59bdd752704377f7390321a3f7521ca746dd08ca6308a3e5d3c06a3dd638dc58","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2020-05-13T19:42:40Z","title_canon_sha256":"3b0c2af856ba2e94297c17443636091a3ccd1e6e469c6a128a3595bfa7b861ee"},"schema_version":"1.0","source":{"id":"2005.06553","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2005.06553","created_at":"2026-07-05T01:11:34Z"},{"alias_kind":"arxiv_version","alias_value":"2005.06553v2","created_at":"2026-07-05T01:11:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2005.06553","created_at":"2026-07-05T01:11:34Z"},{"alias_kind":"pith_short_12","alias_value":"QTE2A5YIWVB4","created_at":"2026-07-05T01:11:34Z"},{"alias_kind":"pith_short_16","alias_value":"QTE2A5YIWVB4FQLX","created_at":"2026-07-05T01:11:34Z"},{"alias_kind":"pith_short_8","alias_value":"QTE2A5YI","created_at":"2026-07-05T01:11:34Z"}],"graph_snapshots":[{"event_id":"sha256:5b857e1d9fffb9fc708e71124358b49518d51ac50f01e2bd49647ed483ab1443","target":"graph","created_at":"2026-07-05T01:11:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2005.06553/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Jascha Sohl-Dickstein","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2020-05-13T19:42:40Z","title":"Two equalities expressing the determinant of a matrix in terms of expectations over matrix-vector products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.06553","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:14b88d781b6e080ff1ff04b6113558499123b4d3a2278f6654d3b14b7834a404","target":"record","created_at":"2026-07-05T01:11:34Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"59bdd752704377f7390321a3f7521ca746dd08ca6308a3e5d3c06a3dd638dc58","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.CO","submitted_at":"2020-05-13T19:42:40Z","title_canon_sha256":"3b0c2af856ba2e94297c17443636091a3ccd1e6e469c6a128a3595bfa7b861ee"},"schema_version":"1.0","source":{"id":"2005.06553","kind":"arxiv","version":2}},"canonical_sha256":"84c9a07708b543c2c1773d47e2367fde3fc46a5921303a271fa350c117f893a0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"84c9a07708b543c2c1773d47e2367fde3fc46a5921303a271fa350c117f893a0","first_computed_at":"2026-07-05T01:11:34.328086Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:11:34.328086Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oQKQlOX+b7oPKgf54FpWUuh/amLkAyMQw2k2XuB+UakuEOfWPxkYzTu9ZGqcCnn6he+dBlNSIQ7XJ+bWCjbtAw==","signature_status":"signed_v1","signed_at":"2026-07-05T01:11:34.328488Z","signed_message":"canonical_sha256_bytes"},"source_id":"2005.06553","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14b88d781b6e080ff1ff04b6113558499123b4d3a2278f6654d3b14b7834a404","sha256:5b857e1d9fffb9fc708e71124358b49518d51ac50f01e2bd49647ed483ab1443"],"state_sha256":"cf637192642f31cc33ef17b82e9c377ddd43dd8f06a24b581b2f2a49119f49d4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i1+RaIZ6wkgOHs4Wd2OUhX3crSmfgd4HStUjdIfCRYBD1pa3mqhb008mptxMvf2Lqp+FBs6X9K5whTqBKTljCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-20T11:54:35.824010Z","bundle_sha256":"f9d79b4a09f2948b399434185f3884e2a832eaa8c841d5204e1d7ba7aff7f19d"}}