{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:UNPKT54ZIWKATVZ5ACLXWMU3SH","short_pith_number":"pith:UNPKT54Z","schema_version":"1.0","canonical_sha256":"a35ea9f799459409d73d00977b329b91fca61c2d0a25d00cd26579d670273dd1","source":{"kind":"arxiv","id":"1611.01540","version":4},"attestation_state":"computed","paper":{"title":"Topology and Geometry of Half-Rectified Network Optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"C. Daniel Freeman, Joan Bruna","submitted_at":"2016-11-04T21:17:42Z","abstract_excerpt":"The loss surface of deep neural networks has recently attracted interest in the optimization and machine learning communities as a prime example of high-dimensional non-convex problem. Some insights were recently gained using spin glass models and mean-field approximations, but at the expense of strongly simplifying the nonlinear nature of the model.\n  In this work, we do not make any such assumption and study conditions on the data distribution and model architecture that prevent the existence of bad local minima. Our theoretical work quantifies and formalizes two important \\emph{folklore} fa"},"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":"1611.01540","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-11-04T21:17:42Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"60eab8a665c5259267c2960f6732820f80c45e8a5469203d34a462411dd1467c","abstract_canon_sha256":"a12f69396dfaa5b26d85a98847e57a5359e6daa760af06952f33e5b6144798dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:13.947970Z","signature_b64":"rZiD8tfDfdnvBqWyj484Ukx+AaPCLVySRLhGmpn7TNUWLZ51/qXMfgJq23ODEa9HIfhrHh8Bvg+487okan76DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a35ea9f799459409d73d00977b329b91fca61c2d0a25d00cd26579d670273dd1","last_reissued_at":"2026-05-18T00:43:13.947359Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:13.947359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Topology and Geometry of Half-Rectified Network Optimization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"C. Daniel Freeman, Joan Bruna","submitted_at":"2016-11-04T21:17:42Z","abstract_excerpt":"The loss surface of deep neural networks has recently attracted interest in the optimization and machine learning communities as a prime example of high-dimensional non-convex problem. Some insights were recently gained using spin glass models and mean-field approximations, but at the expense of strongly simplifying the nonlinear nature of the model.\n  In this work, we do not make any such assumption and study conditions on the data distribution and model architecture that prevent the existence of bad local minima. Our theoretical work quantifies and formalizes two important \\emph{folklore} fa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.01540","kind":"arxiv","version":4},"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":"1611.01540","created_at":"2026-05-18T00:43:13.947461+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.01540v4","created_at":"2026-05-18T00:43:13.947461+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.01540","created_at":"2026-05-18T00:43:13.947461+00:00"},{"alias_kind":"pith_short_12","alias_value":"UNPKT54ZIWKA","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"UNPKT54ZIWKATVZ5","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"UNPKT54Z","created_at":"2026-05-18T12:30:46.583412+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":3,"internal_anchor_count":2,"sample":[{"citing_arxiv_id":"1907.02911","citing_title":"Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss landscape","ref_index":10,"is_internal_anchor":true},{"citing_arxiv_id":"2405.07987","citing_title":"The Platonic Representation Hypothesis","ref_index":52,"is_internal_anchor":true},{"citing_arxiv_id":"2605.09209","citing_title":"Select-then-differentiate: Solving Bilevel Optimization with Manifold Lower-level Solution Sets","ref_index":8,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UNPKT54ZIWKATVZ5ACLXWMU3SH","json":"https://pith.science/pith/UNPKT54ZIWKATVZ5ACLXWMU3SH.json","graph_json":"https://pith.science/api/pith-number/UNPKT54ZIWKATVZ5ACLXWMU3SH/graph.json","events_json":"https://pith.science/api/pith-number/UNPKT54ZIWKATVZ5ACLXWMU3SH/events.json","paper":"https://pith.science/paper/UNPKT54Z"},"agent_actions":{"view_html":"https://pith.science/pith/UNPKT54ZIWKATVZ5ACLXWMU3SH","download_json":"https://pith.science/pith/UNPKT54ZIWKATVZ5ACLXWMU3SH.json","view_paper":"https://pith.science/paper/UNPKT54Z","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.01540&json=true","fetch_graph":"https://pith.science/api/pith-number/UNPKT54ZIWKATVZ5ACLXWMU3SH/graph.json","fetch_events":"https://pith.science/api/pith-number/UNPKT54ZIWKATVZ5ACLXWMU3SH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UNPKT54ZIWKATVZ5ACLXWMU3SH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UNPKT54ZIWKATVZ5ACLXWMU3SH/action/storage_attestation","attest_author":"https://pith.science/pith/UNPKT54ZIWKATVZ5ACLXWMU3SH/action/author_attestation","sign_citation":"https://pith.science/pith/UNPKT54ZIWKATVZ5ACLXWMU3SH/action/citation_signature","submit_replication":"https://pith.science/pith/UNPKT54ZIWKATVZ5ACLXWMU3SH/action/replication_record"}},"created_at":"2026-05-18T00:43:13.947461+00:00","updated_at":"2026-05-18T00:43:13.947461+00:00"}