{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:JULV62LEMTXMT4F7UDNSFGBRFN","short_pith_number":"pith:JULV62LE","schema_version":"1.0","canonical_sha256":"4d175f696464eec9f0bfa0db2298312b41c542312aa3648ba0453a7c10a248d3","source":{"kind":"arxiv","id":"2311.10892","version":1},"attestation_state":"computed","paper":{"title":"The Hidden Linear Structure in Score-Based Models and its Application","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","cs.NA","cs.NE","math.NA","stat.CO"],"primary_cat":"cs.AI","authors_text":"Binxu Wang, John J. Vastola","submitted_at":"2023-11-17T22:25:07Z","abstract_excerpt":"Score-based models have achieved remarkable results in the generative modeling of many domains. By learning the gradient of smoothed data distribution, they can iteratively generate samples from complex distribution e.g. natural images. However, is there any universal structure in the gradient field that will eventually be learned by any neural network? Here, we aim to find such structures through a normative analysis of the score function. First, we derived the closed-form solution to the scored-based model with a Gaussian score. We claimed that for well-trained diffusion models, the learned "},"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":"2311.10892","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.AI","submitted_at":"2023-11-17T22:25:07Z","cross_cats_sorted":["cs.LG","cs.NA","cs.NE","math.NA","stat.CO"],"title_canon_sha256":"7e2c2a86e0188eb90e5596db6e3910ff8811304c4c7174efbd166dd1f32d7698","abstract_canon_sha256":"21fbfa19c0d0ffd76ff336e9f2c6538a418dbf47619eeab213dbcdc237737525"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:00:28.402470Z","signature_b64":"zHA2F02ZRgJh9Ls2zJ4THzQ2iY2xicbm9hsOydqrfyzNlqh2aKx1VLUsYaYUEp9JQoEAxpKbinFPhsEDeIo9Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4d175f696464eec9f0bfa0db2298312b41c542312aa3648ba0453a7c10a248d3","last_reissued_at":"2026-07-05T08:00:28.401990Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:00:28.401990Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Hidden Linear Structure in Score-Based Models and its Application","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","cs.NA","cs.NE","math.NA","stat.CO"],"primary_cat":"cs.AI","authors_text":"Binxu Wang, John J. Vastola","submitted_at":"2023-11-17T22:25:07Z","abstract_excerpt":"Score-based models have achieved remarkable results in the generative modeling of many domains. By learning the gradient of smoothed data distribution, they can iteratively generate samples from complex distribution e.g. natural images. However, is there any universal structure in the gradient field that will eventually be learned by any neural network? Here, we aim to find such structures through a normative analysis of the score function. First, we derived the closed-form solution to the scored-based model with a Gaussian score. We claimed that for well-trained diffusion models, the learned "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2311.10892","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/2311.10892/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":"2311.10892","created_at":"2026-07-05T08:00:28.402047+00:00"},{"alias_kind":"arxiv_version","alias_value":"2311.10892v1","created_at":"2026-07-05T08:00:28.402047+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2311.10892","created_at":"2026-07-05T08:00:28.402047+00:00"},{"alias_kind":"pith_short_12","alias_value":"JULV62LEMTXM","created_at":"2026-07-05T08:00:28.402047+00:00"},{"alias_kind":"pith_short_16","alias_value":"JULV62LEMTXMT4F7","created_at":"2026-07-05T08:00:28.402047+00:00"},{"alias_kind":"pith_short_8","alias_value":"JULV62LE","created_at":"2026-07-05T08:00:28.402047+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":6,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.16415","citing_title":"Diffusion Models, Denoiser Architecture and Creativity","ref_index":14,"is_internal_anchor":false},{"citing_arxiv_id":"2503.11615","citing_title":"From Score Matching to Diffusion: A Fine-Grained Error Analysis in the Gaussian Setting","ref_index":38,"is_internal_anchor":false},{"citing_arxiv_id":"2505.16024","citing_title":"Toward Theoretical Insights into Diffusion Trajectory Distillation via Operator Merging","ref_index":36,"is_internal_anchor":false},{"citing_arxiv_id":"2605.16415","citing_title":"Diffusion Models, Denoiser Architecture and Creativity","ref_index":14,"is_internal_anchor":false},{"citing_arxiv_id":"2605.08392","citing_title":"Geometry-Aware Discretization Error of Diffusion Models","ref_index":13,"is_internal_anchor":false},{"citing_arxiv_id":"2605.10019","citing_title":"The two clocks and the innovation window: When and how generative models learn rules","ref_index":75,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JULV62LEMTXMT4F7UDNSFGBRFN","json":"https://pith.science/pith/JULV62LEMTXMT4F7UDNSFGBRFN.json","graph_json":"https://pith.science/api/pith-number/JULV62LEMTXMT4F7UDNSFGBRFN/graph.json","events_json":"https://pith.science/api/pith-number/JULV62LEMTXMT4F7UDNSFGBRFN/events.json","paper":"https://pith.science/paper/JULV62LE"},"agent_actions":{"view_html":"https://pith.science/pith/JULV62LEMTXMT4F7UDNSFGBRFN","download_json":"https://pith.science/pith/JULV62LEMTXMT4F7UDNSFGBRFN.json","view_paper":"https://pith.science/paper/JULV62LE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2311.10892&json=true","fetch_graph":"https://pith.science/api/pith-number/JULV62LEMTXMT4F7UDNSFGBRFN/graph.json","fetch_events":"https://pith.science/api/pith-number/JULV62LEMTXMT4F7UDNSFGBRFN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JULV62LEMTXMT4F7UDNSFGBRFN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JULV62LEMTXMT4F7UDNSFGBRFN/action/storage_attestation","attest_author":"https://pith.science/pith/JULV62LEMTXMT4F7UDNSFGBRFN/action/author_attestation","sign_citation":"https://pith.science/pith/JULV62LEMTXMT4F7UDNSFGBRFN/action/citation_signature","submit_replication":"https://pith.science/pith/JULV62LEMTXMT4F7UDNSFGBRFN/action/replication_record"}},"created_at":"2026-07-05T08:00:28.402047+00:00","updated_at":"2026-07-05T08:00:28.402047+00:00"}