{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:KSTXNQSM4E7VSIGXE5BLZNID5F","short_pith_number":"pith:KSTXNQSM","schema_version":"1.0","canonical_sha256":"54a776c24ce13f5920d72742bcb503e948981a9dae48b8d6f6541ae9e4d991f5","source":{"kind":"arxiv","id":"2501.18915","version":2},"attestation_state":"computed","paper":{"title":"Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"cs.LG","authors_text":"Giovanni Luca Marchetti, Kathl\\'en Kohn, Matthew Trager, Stefano Mereta, Vahid Shahverdi","submitted_at":"2025-01-31T06:33:58Z","abstract_excerpt":"In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations, whose associated function spaces are semi-algebraic varieties. We outline a dictionary between algebro-geometric invariants of these varieties, such as dimension, degree, and singularities, and fundamental aspects of machine learning, such as sample complexity, expressivity, training dynamics, and implicit bias. Along the way, we review the literature and d"},"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":"2501.18915","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2025-01-31T06:33:58Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"bd142ad2b386c18d4439ce95f6c26ed75440329b6a3e66093f79dd2d08db3bb4","abstract_canon_sha256":"963dffb95213685a95fd8e99b0172fb1e9f2a66202e8f4cc6515649290ac255d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:13:19.594865Z","signature_b64":"hrgNBUhf0TfUnGTTyPudXaRYTLAFD0EOGNk9u/cGxVhqitLmkB//7bM4B9w4rsY87v+3oZWOaYGTDWbicrbIAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"54a776c24ce13f5920d72742bcb503e948981a9dae48b8d6f6541ae9e4d991f5","last_reissued_at":"2026-07-05T11:13:19.594385Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:13:19.594385Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"cs.LG","authors_text":"Giovanni Luca Marchetti, Kathl\\'en Kohn, Matthew Trager, Stefano Mereta, Vahid Shahverdi","submitted_at":"2025-01-31T06:33:58Z","abstract_excerpt":"In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations, whose associated function spaces are semi-algebraic varieties. We outline a dictionary between algebro-geometric invariants of these varieties, such as dimension, degree, and singularities, and fundamental aspects of machine learning, such as sample complexity, expressivity, training dynamics, and implicit bias. Along the way, we review the literature and d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.18915","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/2501.18915/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":"2501.18915","created_at":"2026-07-05T11:13:19.594446+00:00"},{"alias_kind":"arxiv_version","alias_value":"2501.18915v2","created_at":"2026-07-05T11:13:19.594446+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.18915","created_at":"2026-07-05T11:13:19.594446+00:00"},{"alias_kind":"pith_short_12","alias_value":"KSTXNQSM4E7V","created_at":"2026-07-05T11:13:19.594446+00:00"},{"alias_kind":"pith_short_16","alias_value":"KSTXNQSM4E7VSIGX","created_at":"2026-07-05T11:13:19.594446+00:00"},{"alias_kind":"pith_short_8","alias_value":"KSTXNQSM","created_at":"2026-07-05T11:13:19.594446+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":7,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.19268","citing_title":"Patnaik-Pearson intrinsic dimension for internal representations of neural networks","ref_index":20,"is_internal_anchor":false},{"citing_arxiv_id":"2606.19268","citing_title":"Patnaik-Pearson intrinsic dimension for internal representations of neural networks","ref_index":20,"is_internal_anchor":false},{"citing_arxiv_id":"2606.18440","citing_title":"Algebraic Networks and Architectural Degenerations","ref_index":23,"is_internal_anchor":false},{"citing_arxiv_id":"2606.10913","citing_title":"Conservation Laws from Data Symmetry in Neural Networks","ref_index":32,"is_internal_anchor":false},{"citing_arxiv_id":"2605.09609","citing_title":"Minimal Filling Architectures of Polynomial Neural Networks: Counterexamples, Frontier Search, and Defects","ref_index":11,"is_internal_anchor":false},{"citing_arxiv_id":"2605.15970","citing_title":"Copositive Matrices with Ordered Off-Diagonal Entries","ref_index":299,"is_internal_anchor":false},{"citing_arxiv_id":"2605.09609","citing_title":"Minimal Filling Architectures of Polynomial Neural Networks: Counterexamples, Frontier Search, and Defects","ref_index":11,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KSTXNQSM4E7VSIGXE5BLZNID5F","json":"https://pith.science/pith/KSTXNQSM4E7VSIGXE5BLZNID5F.json","graph_json":"https://pith.science/api/pith-number/KSTXNQSM4E7VSIGXE5BLZNID5F/graph.json","events_json":"https://pith.science/api/pith-number/KSTXNQSM4E7VSIGXE5BLZNID5F/events.json","paper":"https://pith.science/paper/KSTXNQSM"},"agent_actions":{"view_html":"https://pith.science/pith/KSTXNQSM4E7VSIGXE5BLZNID5F","download_json":"https://pith.science/pith/KSTXNQSM4E7VSIGXE5BLZNID5F.json","view_paper":"https://pith.science/paper/KSTXNQSM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2501.18915&json=true","fetch_graph":"https://pith.science/api/pith-number/KSTXNQSM4E7VSIGXE5BLZNID5F/graph.json","fetch_events":"https://pith.science/api/pith-number/KSTXNQSM4E7VSIGXE5BLZNID5F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KSTXNQSM4E7VSIGXE5BLZNID5F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KSTXNQSM4E7VSIGXE5BLZNID5F/action/storage_attestation","attest_author":"https://pith.science/pith/KSTXNQSM4E7VSIGXE5BLZNID5F/action/author_attestation","sign_citation":"https://pith.science/pith/KSTXNQSM4E7VSIGXE5BLZNID5F/action/citation_signature","submit_replication":"https://pith.science/pith/KSTXNQSM4E7VSIGXE5BLZNID5F/action/replication_record"}},"created_at":"2026-07-05T11:13:19.594446+00:00","updated_at":"2026-07-05T11:13:19.594446+00:00"}