{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:BEV3NUNCDU2CQCDRWN53FJAGBG","short_pith_number":"pith:BEV3NUNC","schema_version":"1.0","canonical_sha256":"092bb6d1a21d34280871b37bb2a4060999e8ade8185088ba5f0d97b65ccf2ae5","source":{"kind":"arxiv","id":"1711.10455","version":3},"attestation_state":"computed","paper":{"title":"Backprop as Functor: A compositional perspective on supervised learning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.AI","cs.LG"],"primary_cat":"math.CT","authors_text":"Brendan Fong, David I. Spivak, R\\'emy Tuy\\'eras","submitted_at":"2017-11-28T18:34:45Z","abstract_excerpt":"A supervised learning algorithm searches over a set of functions $A \\to B$ parametrised by a space $P$ to find the best approximation to some ideal function $f\\colon A \\to B$. It does this by taking examples $(a,f(a)) \\in A\\times B$, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent---with respect to a fixed step size and an error function satisfying a certain property---defines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural p"},"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":"1711.10455","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2017-11-28T18:34:45Z","cross_cats_sorted":["cs.AI","cs.LG"],"title_canon_sha256":"28e3610d71c56388f626a754ed4961ec890df27fb464544a24e32966e232aca0","abstract_canon_sha256":"442ef518d086ff903d333a607fa0863f019d3577563bffcde6f7387a8e56cf1b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:16.160156Z","signature_b64":"WtWMatpFCR7OrQ9VxOa1kOF+LBnd5bzwO5PYuNTYNk6XO/Rsx37w2t3wzdmazHn6jgTRqTj8AU2zaT4WuMHIBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"092bb6d1a21d34280871b37bb2a4060999e8ade8185088ba5f0d97b65ccf2ae5","last_reissued_at":"2026-05-17T23:47:16.159667Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:16.159667Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Backprop as Functor: A compositional perspective on supervised learning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.AI","cs.LG"],"primary_cat":"math.CT","authors_text":"Brendan Fong, David I. Spivak, R\\'emy Tuy\\'eras","submitted_at":"2017-11-28T18:34:45Z","abstract_excerpt":"A supervised learning algorithm searches over a set of functions $A \\to B$ parametrised by a space $P$ to find the best approximation to some ideal function $f\\colon A \\to B$. It does this by taking examples $(a,f(a)) \\in A\\times B$, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent---with respect to a fixed step size and an error function satisfying a certain property---defines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.10455","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":""},"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":"1711.10455","created_at":"2026-05-17T23:47:16.159746+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.10455v3","created_at":"2026-05-17T23:47:16.159746+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.10455","created_at":"2026-05-17T23:47:16.159746+00:00"},{"alias_kind":"pith_short_12","alias_value":"BEV3NUNCDU2C","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"BEV3NUNCDU2CQCDR","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"BEV3NUNC","created_at":"2026-05-18T12:31:08.081275+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.08934","citing_title":"From Mechanistic to Compositional Interpretability","ref_index":32,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BEV3NUNCDU2CQCDRWN53FJAGBG","json":"https://pith.science/pith/BEV3NUNCDU2CQCDRWN53FJAGBG.json","graph_json":"https://pith.science/api/pith-number/BEV3NUNCDU2CQCDRWN53FJAGBG/graph.json","events_json":"https://pith.science/api/pith-number/BEV3NUNCDU2CQCDRWN53FJAGBG/events.json","paper":"https://pith.science/paper/BEV3NUNC"},"agent_actions":{"view_html":"https://pith.science/pith/BEV3NUNCDU2CQCDRWN53FJAGBG","download_json":"https://pith.science/pith/BEV3NUNCDU2CQCDRWN53FJAGBG.json","view_paper":"https://pith.science/paper/BEV3NUNC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.10455&json=true","fetch_graph":"https://pith.science/api/pith-number/BEV3NUNCDU2CQCDRWN53FJAGBG/graph.json","fetch_events":"https://pith.science/api/pith-number/BEV3NUNCDU2CQCDRWN53FJAGBG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BEV3NUNCDU2CQCDRWN53FJAGBG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BEV3NUNCDU2CQCDRWN53FJAGBG/action/storage_attestation","attest_author":"https://pith.science/pith/BEV3NUNCDU2CQCDRWN53FJAGBG/action/author_attestation","sign_citation":"https://pith.science/pith/BEV3NUNCDU2CQCDRWN53FJAGBG/action/citation_signature","submit_replication":"https://pith.science/pith/BEV3NUNCDU2CQCDRWN53FJAGBG/action/replication_record"}},"created_at":"2026-05-17T23:47:16.159746+00:00","updated_at":"2026-05-17T23:47:16.159746+00:00"}