{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:OUP3553CJMGMOQNID5VOU37KDZ","short_pith_number":"pith:OUP3553C","canonical_record":{"source":{"id":"2105.12806","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-05-26T19:49:47Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"fe7c84a965b15bb4cd9690db08b716778cdde10a78543ac8f6ae7372753df551","abstract_canon_sha256":"7e5e311185dc631c2d0320ccf194243c2ce2a07b5b524923baabf672a64b972e"},"schema_version":"1.0"},"canonical_sha256":"751fbef7624b0cc741a81f6aea6fea1e4aad74f6232e5190c2e0bef91d1eaa21","source":{"kind":"arxiv","id":"2105.12806","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2105.12806","created_at":"2026-07-05T05:28:04Z"},{"alias_kind":"arxiv_version","alias_value":"2105.12806v4","created_at":"2026-07-05T05:28:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.12806","created_at":"2026-07-05T05:28:04Z"},{"alias_kind":"pith_short_12","alias_value":"OUP3553CJMGM","created_at":"2026-07-05T05:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"OUP3553CJMGMOQNI","created_at":"2026-07-05T05:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"OUP3553C","created_at":"2026-07-05T05:28:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:OUP3553CJMGMOQNID5VOU37KDZ","target":"record","payload":{"canonical_record":{"source":{"id":"2105.12806","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-05-26T19:49:47Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"fe7c84a965b15bb4cd9690db08b716778cdde10a78543ac8f6ae7372753df551","abstract_canon_sha256":"7e5e311185dc631c2d0320ccf194243c2ce2a07b5b524923baabf672a64b972e"},"schema_version":"1.0"},"canonical_sha256":"751fbef7624b0cc741a81f6aea6fea1e4aad74f6232e5190c2e0bef91d1eaa21","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:28:04.833013Z","signature_b64":"UJhfzE4fAiDLLl6N7OJw0d3oNO3LT22IylR/Kl0g4sQKZojRu3EyTlhAGTILX+EG1mHx2xYT9/lDM6Wq9rquBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"751fbef7624b0cc741a81f6aea6fea1e4aad74f6232e5190c2e0bef91d1eaa21","last_reissued_at":"2026-07-05T05:28:04.832585Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:28:04.832585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2105.12806","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T05:28:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2Dkm8H0nobrwN8K3BNLx1wcyuN5EbDLUKAKnB+muh4u25FSuVeTtEkXbgOPxqYLIg4SDWSo7+oZXP8yMJUsuCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T00:07:48.004258Z"},"content_sha256":"edb297737036921ef809b3385e367fe942a7e3d9aecea6be89943c5aadf439e6","schema_version":"1.0","event_id":"sha256:edb297737036921ef809b3385e367fe942a7e3d9aecea6be89943c5aadf439e6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:OUP3553CJMGMOQNID5VOU37KDZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Universal Law of Robustness via Isoperimetry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"cs.LG","authors_text":"Mark Sellke, S\\'ebastien Bubeck","submitted_at":"2021-05-26T19:49:47Z","abstract_excerpt":"Classically, data interpolation with a parametrized model class is possible as long as the number of parameters is larger than the number of equations to be satisfied. A puzzling phenomenon in deep learning is that models are trained with many more parameters than what this classical theory would suggest. We propose a partial theoretical explanation for this phenomenon. We prove that for a broad class of data distributions and model classes, overparametrization is necessary if one wants to interpolate the data smoothly. Namely we show that smooth interpolation requires $d$ times more parameter"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.12806","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2105.12806/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":"2026-07-05T05:28:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aiH0oq4/a5UKfnlN7xUzcDwzLSUmXZ9kolZPhkgV/neh8TiRPwQqMaq4heaUCT9yODG9PUcpiFnp0Cg3GFRWBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-07T00:07:48.004777Z"},"content_sha256":"bada4e4140bb5075f4bed2375fd4735bd0ac2f5ba76da4208a57f7716cccb36d","schema_version":"1.0","event_id":"sha256:bada4e4140bb5075f4bed2375fd4735bd0ac2f5ba76da4208a57f7716cccb36d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OUP3553CJMGMOQNID5VOU37KDZ/bundle.json","state_url":"https://pith.science/pith/OUP3553CJMGMOQNID5VOU37KDZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OUP3553CJMGMOQNID5VOU37KDZ/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":"2026-08-07T00:07:48Z","links":{"resolver":"https://pith.science/pith/OUP3553CJMGMOQNID5VOU37KDZ","bundle":"https://pith.science/pith/OUP3553CJMGMOQNID5VOU37KDZ/bundle.json","state":"https://pith.science/pith/OUP3553CJMGMOQNID5VOU37KDZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OUP3553CJMGMOQNID5VOU37KDZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:OUP3553CJMGMOQNID5VOU37KDZ","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":"7e5e311185dc631c2d0320ccf194243c2ce2a07b5b524923baabf672a64b972e","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-05-26T19:49:47Z","title_canon_sha256":"fe7c84a965b15bb4cd9690db08b716778cdde10a78543ac8f6ae7372753df551"},"schema_version":"1.0","source":{"id":"2105.12806","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2105.12806","created_at":"2026-07-05T05:28:04Z"},{"alias_kind":"arxiv_version","alias_value":"2105.12806v4","created_at":"2026-07-05T05:28:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.12806","created_at":"2026-07-05T05:28:04Z"},{"alias_kind":"pith_short_12","alias_value":"OUP3553CJMGM","created_at":"2026-07-05T05:28:04Z"},{"alias_kind":"pith_short_16","alias_value":"OUP3553CJMGMOQNI","created_at":"2026-07-05T05:28:04Z"},{"alias_kind":"pith_short_8","alias_value":"OUP3553C","created_at":"2026-07-05T05:28:04Z"}],"graph_snapshots":[{"event_id":"sha256:bada4e4140bb5075f4bed2375fd4735bd0ac2f5ba76da4208a57f7716cccb36d","target":"graph","created_at":"2026-07-05T05:28:04Z","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/2105.12806/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Classically, data interpolation with a parametrized model class is possible as long as the number of parameters is larger than the number of equations to be satisfied. A puzzling phenomenon in deep learning is that models are trained with many more parameters than what this classical theory would suggest. We propose a partial theoretical explanation for this phenomenon. We prove that for a broad class of data distributions and model classes, overparametrization is necessary if one wants to interpolate the data smoothly. Namely we show that smooth interpolation requires $d$ times more parameter","authors_text":"Mark Sellke, S\\'ebastien Bubeck","cross_cats":["stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-05-26T19:49:47Z","title":"A Universal Law of Robustness via Isoperimetry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.12806","kind":"arxiv","version":4},"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:edb297737036921ef809b3385e367fe942a7e3d9aecea6be89943c5aadf439e6","target":"record","created_at":"2026-07-05T05:28:04Z","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":"7e5e311185dc631c2d0320ccf194243c2ce2a07b5b524923baabf672a64b972e","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2021-05-26T19:49:47Z","title_canon_sha256":"fe7c84a965b15bb4cd9690db08b716778cdde10a78543ac8f6ae7372753df551"},"schema_version":"1.0","source":{"id":"2105.12806","kind":"arxiv","version":4}},"canonical_sha256":"751fbef7624b0cc741a81f6aea6fea1e4aad74f6232e5190c2e0bef91d1eaa21","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"751fbef7624b0cc741a81f6aea6fea1e4aad74f6232e5190c2e0bef91d1eaa21","first_computed_at":"2026-07-05T05:28:04.832585Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:28:04.832585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UJhfzE4fAiDLLl6N7OJw0d3oNO3LT22IylR/Kl0g4sQKZojRu3EyTlhAGTILX+EG1mHx2xYT9/lDM6Wq9rquBg==","signature_status":"signed_v1","signed_at":"2026-07-05T05:28:04.833013Z","signed_message":"canonical_sha256_bytes"},"source_id":"2105.12806","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:edb297737036921ef809b3385e367fe942a7e3d9aecea6be89943c5aadf439e6","sha256:bada4e4140bb5075f4bed2375fd4735bd0ac2f5ba76da4208a57f7716cccb36d"],"state_sha256":"929a0020f2a3f6b9c22bd1056f0aa38a5e2aa8a1cb1a0912160fe70ad5172b16"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vGCqNgYY8fZcnz/DrvVzGVw8/VGFrINT+xAB8CVg8K4du9MVkIdak778up0/1aepApK3kI2kpNcHThUR7xbmBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-07T00:07:48.037703Z","bundle_sha256":"48ecb97bc9d043256aae5dc27153a882eb5c9f36220c1f72fcd1a697a4f59ce6"}}