{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:Z5Y4YYH33PE2AFGKKHOVHGMQBA","short_pith_number":"pith:Z5Y4YYH3","schema_version":"1.0","canonical_sha256":"cf71cc60fbdbc9a014ca51dd5399900823f0abfe4efe4f6a3d50d3e21cde1e6c","source":{"kind":"arxiv","id":"2412.07904","version":3},"attestation_state":"computed","paper":{"title":"Score Change of Variables","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI","math.PR"],"primary_cat":"cs.LG","authors_text":"Stephen Robbins","submitted_at":"2024-12-10T20:27:15Z","abstract_excerpt":"We derive a general change of variables formula for score functions, showing that for a smooth, invertible transformation $\\mathbf{y} = \\phi(\\mathbf{x})$, the transformed score function $\\nabla_{\\mathbf{y}} \\log q(\\mathbf{y})$ can be expressed directly in terms of $\\nabla_{\\mathbf{x}} \\log p(\\mathbf{x})$. Using this result, we develop two applications: First, we establish a reverse-time It\\^o lemma for score-based diffusion models, allowing the use of $\\nabla_{\\mathbf{x}} \\log p_t(\\mathbf{x})$ to reverse an SDE in the transformed space without directly learning $\\nabla_{\\mathbf{y}} \\log q_t(\\m"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2412.07904","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-12-10T20:27:15Z","cross_cats_sorted":["cs.AI","math.PR"],"title_canon_sha256":"06795a68a7c03ce73efe7a7802e0e0e79ea1dd73ed4b000f88c16b7faca3d59d","abstract_canon_sha256":"aaeae34bcf024b52168d513c79ee758047b98913433cc0a57313c82b765fb095"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:18:46.625657Z","signature_b64":"xYo0Pb1l9OFFQKIpWdM7v+qxnIfrpjGhYyjdzK0QLnwDCKa6z79NL2fs58v2fODlHfNcsYuL0R7R/vZDQ+tpBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf71cc60fbdbc9a014ca51dd5399900823f0abfe4efe4f6a3d50d3e21cde1e6c","last_reissued_at":"2026-07-05T10:18:46.625051Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:18:46.625051Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Score Change of Variables","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI","math.PR"],"primary_cat":"cs.LG","authors_text":"Stephen Robbins","submitted_at":"2024-12-10T20:27:15Z","abstract_excerpt":"We derive a general change of variables formula for score functions, showing that for a smooth, invertible transformation $\\mathbf{y} = \\phi(\\mathbf{x})$, the transformed score function $\\nabla_{\\mathbf{y}} \\log q(\\mathbf{y})$ can be expressed directly in terms of $\\nabla_{\\mathbf{x}} \\log p(\\mathbf{x})$. Using this result, we develop two applications: First, we establish a reverse-time It\\^o lemma for score-based diffusion models, allowing the use of $\\nabla_{\\mathbf{x}} \\log p_t(\\mathbf{x})$ to reverse an SDE in the transformed space without directly learning $\\nabla_{\\mathbf{y}} \\log q_t(\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.07904","kind":"arxiv","version":3},"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/2412.07904/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2412.07904","created_at":"2026-07-05T10:18:46.625128+00:00"},{"alias_kind":"arxiv_version","alias_value":"2412.07904v3","created_at":"2026-07-05T10:18:46.625128+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.07904","created_at":"2026-07-05T10:18:46.625128+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z5Y4YYH33PE2","created_at":"2026-07-05T10:18:46.625128+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z5Y4YYH33PE2AFGK","created_at":"2026-07-05T10:18:46.625128+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z5Y4YYH3","created_at":"2026-07-05T10:18:46.625128+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/Z5Y4YYH33PE2AFGKKHOVHGMQBA","json":"https://pith.science/pith/Z5Y4YYH33PE2AFGKKHOVHGMQBA.json","graph_json":"https://pith.science/api/pith-number/Z5Y4YYH33PE2AFGKKHOVHGMQBA/graph.json","events_json":"https://pith.science/api/pith-number/Z5Y4YYH33PE2AFGKKHOVHGMQBA/events.json","paper":"https://pith.science/paper/Z5Y4YYH3"},"agent_actions":{"view_html":"https://pith.science/pith/Z5Y4YYH33PE2AFGKKHOVHGMQBA","download_json":"https://pith.science/pith/Z5Y4YYH33PE2AFGKKHOVHGMQBA.json","view_paper":"https://pith.science/paper/Z5Y4YYH3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2412.07904&json=true","fetch_graph":"https://pith.science/api/pith-number/Z5Y4YYH33PE2AFGKKHOVHGMQBA/graph.json","fetch_events":"https://pith.science/api/pith-number/Z5Y4YYH33PE2AFGKKHOVHGMQBA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z5Y4YYH33PE2AFGKKHOVHGMQBA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z5Y4YYH33PE2AFGKKHOVHGMQBA/action/storage_attestation","attest_author":"https://pith.science/pith/Z5Y4YYH33PE2AFGKKHOVHGMQBA/action/author_attestation","sign_citation":"https://pith.science/pith/Z5Y4YYH33PE2AFGKKHOVHGMQBA/action/citation_signature","submit_replication":"https://pith.science/pith/Z5Y4YYH33PE2AFGKKHOVHGMQBA/action/replication_record"}},"created_at":"2026-07-05T10:18:46.625128+00:00","updated_at":"2026-07-05T10:18:46.625128+00:00"}