{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:57VM3GD3PIO5IMJ66W476Z3GEZ","short_pith_number":"pith:57VM3GD3","canonical_record":{"source":{"id":"2411.08798","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-11-13T17:25:25Z","cross_cats_sorted":["cs.NE","math.AG"],"title_canon_sha256":"b2422a18c7a40e919744a2ad728ab82dac7d99fa69549b09f9109a4c55cd440a","abstract_canon_sha256":"f520dd461ce59495fc2ebc417ef070736104878e2ac89ec3b5d9a6689b74e77a"},"schema_version":"1.0"},"canonical_sha256":"efeacd987b7a1dd4313ef5b9ff6766267af3f03cc4e0702570c2ef8016267ca2","source":{"kind":"arxiv","id":"2411.08798","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.08798","created_at":"2026-07-05T10:28:26Z"},{"alias_kind":"arxiv_version","alias_value":"2411.08798v2","created_at":"2026-07-05T10:28:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.08798","created_at":"2026-07-05T10:28:26Z"},{"alias_kind":"pith_short_12","alias_value":"57VM3GD3PIO5","created_at":"2026-07-05T10:28:26Z"},{"alias_kind":"pith_short_16","alias_value":"57VM3GD3PIO5IMJ6","created_at":"2026-07-05T10:28:26Z"},{"alias_kind":"pith_short_8","alias_value":"57VM3GD3","created_at":"2026-07-05T10:28:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:57VM3GD3PIO5IMJ66W476Z3GEZ","target":"record","payload":{"canonical_record":{"source":{"id":"2411.08798","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-11-13T17:25:25Z","cross_cats_sorted":["cs.NE","math.AG"],"title_canon_sha256":"b2422a18c7a40e919744a2ad728ab82dac7d99fa69549b09f9109a4c55cd440a","abstract_canon_sha256":"f520dd461ce59495fc2ebc417ef070736104878e2ac89ec3b5d9a6689b74e77a"},"schema_version":"1.0"},"canonical_sha256":"efeacd987b7a1dd4313ef5b9ff6766267af3f03cc4e0702570c2ef8016267ca2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:28:26.209289Z","signature_b64":"JluWZaCeOYCrvtwyNnClBfdenGcooCxy5k/foylo8USUewYOy9oHQDCoOlxd3aZx+lkNw5tqqo59AGVqEkEICQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"efeacd987b7a1dd4313ef5b9ff6766267af3f03cc4e0702570c2ef8016267ca2","last_reissued_at":"2026-07-05T10:28:26.208733Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:28:26.208733Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.08798","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:28:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QoTLv2EkQ6BZe1/89raGZYLkcQZsYEk0La70tYC+2Rba+Ad9GdUjwfW1fnxnwJtaZSabPMh3jZcTmtPioe/dAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T23:12:04.879659Z"},"content_sha256":"ea897e92e7c587f593741a5b8cafd668c570cda21ce05c41ecc1183e9208c9b4","schema_version":"1.0","event_id":"sha256:ea897e92e7c587f593741a5b8cafd668c570cda21ce05c41ecc1183e9208c9b4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:57VM3GD3PIO5IMJ66W476Z3GEZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Learning Gaussian Multi-Index Models with Gradient Flow: Time Complexity and Directional Convergence","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NE","math.AG"],"primary_cat":"cs.LG","authors_text":"Amire Bendjeddou, Berfin \\c{S}im\\c{s}ek, Daniel Hsu","submitted_at":"2024-11-13T17:25:25Z","abstract_excerpt":"This work focuses on the gradient flow dynamics of a neural network model that uses correlation loss to approximate a multi-index function on high-dimensional standard Gaussian data. Specifically, the multi-index function we consider is a sum of neurons $f^*(x) \\!=\\! \\sum_{j=1}^k \\! \\sigma^*(v_j^T x)$ where $v_1, \\dots, v_k$ are unit vectors, and $\\sigma^*$ lacks the first and second Hermite polynomials in its Hermite expansion. It is known that, for the single-index case ($k\\!=\\!1$), overcoming the search phase requires polynomial time complexity. We first generalize this result to multi-inde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.08798","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/2411.08798/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:28:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JqSWSxrth/NQbkEvBdnpVRuOHJs2t9F/6Vmzm9ZFZ+DljnJtYGxYtq1NjN/t10+4wv7QO3hWFo1/MjcC4h8aCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T23:12:04.880893Z"},"content_sha256":"1e3a4e1c583e776931cd62d7997606d4272fafc70a5cc5b7e9aa3c8a2cfae0fb","schema_version":"1.0","event_id":"sha256:1e3a4e1c583e776931cd62d7997606d4272fafc70a5cc5b7e9aa3c8a2cfae0fb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/57VM3GD3PIO5IMJ66W476Z3GEZ/bundle.json","state_url":"https://pith.science/pith/57VM3GD3PIO5IMJ66W476Z3GEZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/57VM3GD3PIO5IMJ66W476Z3GEZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-10T23:12:04Z","links":{"resolver":"https://pith.science/pith/57VM3GD3PIO5IMJ66W476Z3GEZ","bundle":"https://pith.science/pith/57VM3GD3PIO5IMJ66W476Z3GEZ/bundle.json","state":"https://pith.science/pith/57VM3GD3PIO5IMJ66W476Z3GEZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/57VM3GD3PIO5IMJ66W476Z3GEZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:57VM3GD3PIO5IMJ66W476Z3GEZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f520dd461ce59495fc2ebc417ef070736104878e2ac89ec3b5d9a6689b74e77a","cross_cats_sorted":["cs.NE","math.AG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-11-13T17:25:25Z","title_canon_sha256":"b2422a18c7a40e919744a2ad728ab82dac7d99fa69549b09f9109a4c55cd440a"},"schema_version":"1.0","source":{"id":"2411.08798","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.08798","created_at":"2026-07-05T10:28:26Z"},{"alias_kind":"arxiv_version","alias_value":"2411.08798v2","created_at":"2026-07-05T10:28:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.08798","created_at":"2026-07-05T10:28:26Z"},{"alias_kind":"pith_short_12","alias_value":"57VM3GD3PIO5","created_at":"2026-07-05T10:28:26Z"},{"alias_kind":"pith_short_16","alias_value":"57VM3GD3PIO5IMJ6","created_at":"2026-07-05T10:28:26Z"},{"alias_kind":"pith_short_8","alias_value":"57VM3GD3","created_at":"2026-07-05T10:28:26Z"}],"graph_snapshots":[{"event_id":"sha256:1e3a4e1c583e776931cd62d7997606d4272fafc70a5cc5b7e9aa3c8a2cfae0fb","target":"graph","created_at":"2026-07-05T10:28:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2411.08798/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This work focuses on the gradient flow dynamics of a neural network model that uses correlation loss to approximate a multi-index function on high-dimensional standard Gaussian data. Specifically, the multi-index function we consider is a sum of neurons $f^*(x) \\!=\\! \\sum_{j=1}^k \\! \\sigma^*(v_j^T x)$ where $v_1, \\dots, v_k$ are unit vectors, and $\\sigma^*$ lacks the first and second Hermite polynomials in its Hermite expansion. It is known that, for the single-index case ($k\\!=\\!1$), overcoming the search phase requires polynomial time complexity. We first generalize this result to multi-inde","authors_text":"Amire Bendjeddou, Berfin \\c{S}im\\c{s}ek, Daniel Hsu","cross_cats":["cs.NE","math.AG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-11-13T17:25:25Z","title":"Learning Gaussian Multi-Index Models with Gradient Flow: Time Complexity and Directional Convergence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.08798","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ea897e92e7c587f593741a5b8cafd668c570cda21ce05c41ecc1183e9208c9b4","target":"record","created_at":"2026-07-05T10:28:26Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f520dd461ce59495fc2ebc417ef070736104878e2ac89ec3b5d9a6689b74e77a","cross_cats_sorted":["cs.NE","math.AG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2024-11-13T17:25:25Z","title_canon_sha256":"b2422a18c7a40e919744a2ad728ab82dac7d99fa69549b09f9109a4c55cd440a"},"schema_version":"1.0","source":{"id":"2411.08798","kind":"arxiv","version":2}},"canonical_sha256":"efeacd987b7a1dd4313ef5b9ff6766267af3f03cc4e0702570c2ef8016267ca2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"efeacd987b7a1dd4313ef5b9ff6766267af3f03cc4e0702570c2ef8016267ca2","first_computed_at":"2026-07-05T10:28:26.208733Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:28:26.208733Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JluWZaCeOYCrvtwyNnClBfdenGcooCxy5k/foylo8USUewYOy9oHQDCoOlxd3aZx+lkNw5tqqo59AGVqEkEICQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:28:26.209289Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.08798","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea897e92e7c587f593741a5b8cafd668c570cda21ce05c41ecc1183e9208c9b4","sha256:1e3a4e1c583e776931cd62d7997606d4272fafc70a5cc5b7e9aa3c8a2cfae0fb"],"state_sha256":"09e096d65c035b4505a428c873300b622f6170b27dbe30ccfe8ff68b463de025"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l/sAck2EyEiwGlbXlHSNDT8izNx/qci19xYho0TvQ/VNvq+QCNITgU47v36llXmsfXAC/ZSY0nneDBGLy7e6BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T23:12:04.886152Z","bundle_sha256":"5cf8e91fd7329bc639fad3a9ee8921516c574f5d6ba1fff326ee4ef512504cea"}}