{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:HIBVE3TEUUAQEHXMQ4FINKIEFE","short_pith_number":"pith:HIBVE3TE","schema_version":"1.0","canonical_sha256":"3a03526e64a501021eec870a86a9042913bd5a583d759e289724993286767fb0","source":{"kind":"arxiv","id":"2012.04030","version":2},"attestation_state":"computed","paper":{"title":"Statistical Mechanics of Deep Linear Neural Networks: The Back-Propagating Kernel Renormalization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["physics.app-ph"],"primary_cat":"cs.LG","authors_text":"Haim Sompolinsky, Qianyi Li","submitted_at":"2020-12-07T20:08:31Z","abstract_excerpt":"The success of deep learning in many real-world tasks has triggered an intense effort to understand the power and limitations of deep learning in the training and generalization of complex tasks, so far with limited progress. In this work, we study the statistical mechanics of learning in Deep Linear Neural Networks (DLNNs) in which the input-output function of an individual unit is linear. Despite the linearity of the units, learning in DLNNs is nonlinear, hence studying its properties reveals some of the features of nonlinear Deep Neural Networks (DNNs). Importantly, we solve exactly the net"},"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":"2012.04030","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2020-12-07T20:08:31Z","cross_cats_sorted":["physics.app-ph"],"title_canon_sha256":"bc14ea770c419213eb9a8fec91b7095b8fd0496f616b964d0764de0646dc8542","abstract_canon_sha256":"81544439b146d95f0712dd3714e2752299e7e84c80ae244113b73bc319635037"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:16:07.159609Z","signature_b64":"SVnB34t6ddgaMjPa/LgZ60W9Dghig1uTRo+A3d8XUG5WaUO3Z9qnYnhXud4Keh04VibvsgEW5xm+ryUNalXfBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a03526e64a501021eec870a86a9042913bd5a583d759e289724993286767fb0","last_reissued_at":"2026-07-05T03:16:07.159128Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:16:07.159128Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Statistical Mechanics of Deep Linear Neural Networks: The Back-Propagating Kernel Renormalization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["physics.app-ph"],"primary_cat":"cs.LG","authors_text":"Haim Sompolinsky, Qianyi Li","submitted_at":"2020-12-07T20:08:31Z","abstract_excerpt":"The success of deep learning in many real-world tasks has triggered an intense effort to understand the power and limitations of deep learning in the training and generalization of complex tasks, so far with limited progress. In this work, we study the statistical mechanics of learning in Deep Linear Neural Networks (DLNNs) in which the input-output function of an individual unit is linear. Despite the linearity of the units, learning in DLNNs is nonlinear, hence studying its properties reveals some of the features of nonlinear Deep Neural Networks (DNNs). Importantly, we solve exactly the net"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2012.04030","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/2012.04030/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":"2012.04030","created_at":"2026-07-05T03:16:07.159184+00:00"},{"alias_kind":"arxiv_version","alias_value":"2012.04030v2","created_at":"2026-07-05T03:16:07.159184+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2012.04030","created_at":"2026-07-05T03:16:07.159184+00:00"},{"alias_kind":"pith_short_12","alias_value":"HIBVE3TEUUAQ","created_at":"2026-07-05T03:16:07.159184+00:00"},{"alias_kind":"pith_short_16","alias_value":"HIBVE3TEUUAQEHXM","created_at":"2026-07-05T03:16:07.159184+00:00"},{"alias_kind":"pith_short_8","alias_value":"HIBVE3TE","created_at":"2026-07-05T03:16:07.159184+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2501.11773","citing_title":"Can Bayesian Neural Networks Make Confident Predictions?","ref_index":41,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HIBVE3TEUUAQEHXMQ4FINKIEFE","json":"https://pith.science/pith/HIBVE3TEUUAQEHXMQ4FINKIEFE.json","graph_json":"https://pith.science/api/pith-number/HIBVE3TEUUAQEHXMQ4FINKIEFE/graph.json","events_json":"https://pith.science/api/pith-number/HIBVE3TEUUAQEHXMQ4FINKIEFE/events.json","paper":"https://pith.science/paper/HIBVE3TE"},"agent_actions":{"view_html":"https://pith.science/pith/HIBVE3TEUUAQEHXMQ4FINKIEFE","download_json":"https://pith.science/pith/HIBVE3TEUUAQEHXMQ4FINKIEFE.json","view_paper":"https://pith.science/paper/HIBVE3TE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2012.04030&json=true","fetch_graph":"https://pith.science/api/pith-number/HIBVE3TEUUAQEHXMQ4FINKIEFE/graph.json","fetch_events":"https://pith.science/api/pith-number/HIBVE3TEUUAQEHXMQ4FINKIEFE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HIBVE3TEUUAQEHXMQ4FINKIEFE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HIBVE3TEUUAQEHXMQ4FINKIEFE/action/storage_attestation","attest_author":"https://pith.science/pith/HIBVE3TEUUAQEHXMQ4FINKIEFE/action/author_attestation","sign_citation":"https://pith.science/pith/HIBVE3TEUUAQEHXMQ4FINKIEFE/action/citation_signature","submit_replication":"https://pith.science/pith/HIBVE3TEUUAQEHXMQ4FINKIEFE/action/replication_record"}},"created_at":"2026-07-05T03:16:07.159184+00:00","updated_at":"2026-07-05T03:16:07.159184+00:00"}