{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:57CKEBU7QIQLYUKWNZA277XSKW","short_pith_number":"pith:57CKEBU7","schema_version":"1.0","canonical_sha256":"efc4a2069f8220bc51566e41affef2559fe5c7497898415597ff97cb6d4f5026","source":{"kind":"arxiv","id":"2201.11968","version":2},"attestation_state":"computed","paper":{"title":"Training invariances and the low-rank phenomenon: beyond linear networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"cs.LG","authors_text":"Stefanie Jegelka, Thien Le","submitted_at":"2022-01-28T07:31:19Z","abstract_excerpt":"The implicit bias induced by the training of neural networks has become a topic of rigorous study. In the limit of gradient flow and gradient descent with appropriate step size, it has been shown that when one trains a deep linear network with logistic or exponential loss on linearly separable data, the weights converge to rank-1 matrices. In this paper, we extend this theoretical result to the last few linear layers of the much wider class of nonlinear ReLU-activated feedforward networks containing fully-connected layers and skip connections. Similar to the linear case, the proof relies on sp"},"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":"2201.11968","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2022-01-28T07:31:19Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"921901c049deebc92e3df1895102f6de702a764461cc615d1d14cc751ac4bf8a","abstract_canon_sha256":"9115ba3ba2b89ebaa24cdafefe40ceb4272988ba27720f8231e663108298be30"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:17:38.974697Z","signature_b64":"eTw77Ux5z99fhIFoA+eIWvS9tJEjfcZvlhdf17hcFePPc8dgdLV/kXYGmLtEl9fkt3LtqNQi8hORNxEfZwZDDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"efc4a2069f8220bc51566e41affef2559fe5c7497898415597ff97cb6d4f5026","last_reissued_at":"2026-07-05T04:17:38.974151Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:17:38.974151Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Training invariances and the low-rank phenomenon: beyond linear networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"cs.LG","authors_text":"Stefanie Jegelka, Thien Le","submitted_at":"2022-01-28T07:31:19Z","abstract_excerpt":"The implicit bias induced by the training of neural networks has become a topic of rigorous study. In the limit of gradient flow and gradient descent with appropriate step size, it has been shown that when one trains a deep linear network with logistic or exponential loss on linearly separable data, the weights converge to rank-1 matrices. In this paper, we extend this theoretical result to the last few linear layers of the much wider class of nonlinear ReLU-activated feedforward networks containing fully-connected layers and skip connections. Similar to the linear case, the proof relies on sp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.11968","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/2201.11968/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":"2201.11968","created_at":"2026-07-05T04:17:38.974214+00:00"},{"alias_kind":"arxiv_version","alias_value":"2201.11968v2","created_at":"2026-07-05T04:17:38.974214+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.11968","created_at":"2026-07-05T04:17:38.974214+00:00"},{"alias_kind":"pith_short_12","alias_value":"57CKEBU7QIQL","created_at":"2026-07-05T04:17:38.974214+00:00"},{"alias_kind":"pith_short_16","alias_value":"57CKEBU7QIQLYUKW","created_at":"2026-07-05T04:17:38.974214+00:00"},{"alias_kind":"pith_short_8","alias_value":"57CKEBU7","created_at":"2026-07-05T04:17:38.974214+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2502.05668","citing_title":"The late-stage training dynamics of (stochastic) subgradient descent on homogeneous neural networks","ref_index":34,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/57CKEBU7QIQLYUKWNZA277XSKW","json":"https://pith.science/pith/57CKEBU7QIQLYUKWNZA277XSKW.json","graph_json":"https://pith.science/api/pith-number/57CKEBU7QIQLYUKWNZA277XSKW/graph.json","events_json":"https://pith.science/api/pith-number/57CKEBU7QIQLYUKWNZA277XSKW/events.json","paper":"https://pith.science/paper/57CKEBU7"},"agent_actions":{"view_html":"https://pith.science/pith/57CKEBU7QIQLYUKWNZA277XSKW","download_json":"https://pith.science/pith/57CKEBU7QIQLYUKWNZA277XSKW.json","view_paper":"https://pith.science/paper/57CKEBU7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2201.11968&json=true","fetch_graph":"https://pith.science/api/pith-number/57CKEBU7QIQLYUKWNZA277XSKW/graph.json","fetch_events":"https://pith.science/api/pith-number/57CKEBU7QIQLYUKWNZA277XSKW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/57CKEBU7QIQLYUKWNZA277XSKW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/57CKEBU7QIQLYUKWNZA277XSKW/action/storage_attestation","attest_author":"https://pith.science/pith/57CKEBU7QIQLYUKWNZA277XSKW/action/author_attestation","sign_citation":"https://pith.science/pith/57CKEBU7QIQLYUKWNZA277XSKW/action/citation_signature","submit_replication":"https://pith.science/pith/57CKEBU7QIQLYUKWNZA277XSKW/action/replication_record"}},"created_at":"2026-07-05T04:17:38.974214+00:00","updated_at":"2026-07-05T04:17:38.974214+00:00"}