{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:BYBDUENU3YLGMG6OGHRK2HMV3K","short_pith_number":"pith:BYBDUENU","schema_version":"1.0","canonical_sha256":"0e023a11b4de16661bce31e2ad1d95da82850689025b63e9521700a4700c316b","source":{"kind":"arxiv","id":"2412.00373","version":1},"attestation_state":"computed","paper":{"title":"Approximate Fiber Product: A Preliminary Algebraic-Geometric Perspective on Multimodal Embedding Alignment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.AI","math.AG"],"primary_cat":"cs.LG","authors_text":"Dongfang Zhao","submitted_at":"2024-11-30T06:45:13Z","abstract_excerpt":"Multimodal tasks, such as image-text retrieval and generation, require embedding data from diverse modalities into a shared representation space. Aligning embeddings from heterogeneous sources while preserving shared and modality-specific information is a fundamental challenge. This paper provides an initial attempt to integrate algebraic geometry into multimodal representation learning, offering a foundational perspective for further exploration.\n  We model image and text data as polynomials over discrete rings, \\( \\mathbb{Z}_{256}[x] \\) and \\( \\mathbb{Z}_{|V|}[x] \\), respectively, enabling t"},"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":"2412.00373","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2024-11-30T06:45:13Z","cross_cats_sorted":["cs.AI","math.AG"],"title_canon_sha256":"3d687e8653dce43a806cc3908adcbeebf2cc5f9e70309558d483e5aa2ed0e254","abstract_canon_sha256":"949f13d2328196f6c98f4812604ffc78743538f4adf891aaae387790612b7e57"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:42:40.308310Z","signature_b64":"pz/VVPueKuhZAAr+MERdkyWbYKWePe9bj1L0NtONLylOicBqCphRO0byamPuL74RRX3XZ+inYahOp36Hxg0zAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e023a11b4de16661bce31e2ad1d95da82850689025b63e9521700a4700c316b","last_reissued_at":"2026-07-05T09:42:40.307837Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:42:40.307837Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Approximate Fiber Product: A Preliminary Algebraic-Geometric Perspective on Multimodal Embedding Alignment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.AI","math.AG"],"primary_cat":"cs.LG","authors_text":"Dongfang Zhao","submitted_at":"2024-11-30T06:45:13Z","abstract_excerpt":"Multimodal tasks, such as image-text retrieval and generation, require embedding data from diverse modalities into a shared representation space. Aligning embeddings from heterogeneous sources while preserving shared and modality-specific information is a fundamental challenge. This paper provides an initial attempt to integrate algebraic geometry into multimodal representation learning, offering a foundational perspective for further exploration.\n  We model image and text data as polynomials over discrete rings, \\( \\mathbb{Z}_{256}[x] \\) and \\( \\mathbb{Z}_{|V|}[x] \\), respectively, enabling t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.00373","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/2412.00373/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":"2412.00373","created_at":"2026-07-05T09:42:40.307897+00:00"},{"alias_kind":"arxiv_version","alias_value":"2412.00373v1","created_at":"2026-07-05T09:42:40.307897+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.00373","created_at":"2026-07-05T09:42:40.307897+00:00"},{"alias_kind":"pith_short_12","alias_value":"BYBDUENU3YLG","created_at":"2026-07-05T09:42:40.307897+00:00"},{"alias_kind":"pith_short_16","alias_value":"BYBDUENU3YLGMG6O","created_at":"2026-07-05T09:42:40.307897+00:00"},{"alias_kind":"pith_short_8","alias_value":"BYBDUENU","created_at":"2026-07-05T09:42:40.307897+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/BYBDUENU3YLGMG6OGHRK2HMV3K","json":"https://pith.science/pith/BYBDUENU3YLGMG6OGHRK2HMV3K.json","graph_json":"https://pith.science/api/pith-number/BYBDUENU3YLGMG6OGHRK2HMV3K/graph.json","events_json":"https://pith.science/api/pith-number/BYBDUENU3YLGMG6OGHRK2HMV3K/events.json","paper":"https://pith.science/paper/BYBDUENU"},"agent_actions":{"view_html":"https://pith.science/pith/BYBDUENU3YLGMG6OGHRK2HMV3K","download_json":"https://pith.science/pith/BYBDUENU3YLGMG6OGHRK2HMV3K.json","view_paper":"https://pith.science/paper/BYBDUENU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2412.00373&json=true","fetch_graph":"https://pith.science/api/pith-number/BYBDUENU3YLGMG6OGHRK2HMV3K/graph.json","fetch_events":"https://pith.science/api/pith-number/BYBDUENU3YLGMG6OGHRK2HMV3K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BYBDUENU3YLGMG6OGHRK2HMV3K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BYBDUENU3YLGMG6OGHRK2HMV3K/action/storage_attestation","attest_author":"https://pith.science/pith/BYBDUENU3YLGMG6OGHRK2HMV3K/action/author_attestation","sign_citation":"https://pith.science/pith/BYBDUENU3YLGMG6OGHRK2HMV3K/action/citation_signature","submit_replication":"https://pith.science/pith/BYBDUENU3YLGMG6OGHRK2HMV3K/action/replication_record"}},"created_at":"2026-07-05T09:42:40.307897+00:00","updated_at":"2026-07-05T09:42:40.307897+00:00"}