{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:64FYWNEJNIH5IPEILQ3B236J3V","short_pith_number":"pith:64FYWNEJ","canonical_record":{"source":{"id":"2605.14927","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-14T15:02:24Z","cross_cats_sorted":[],"title_canon_sha256":"2021f781bd8f345eb8a7ac62f5d6ce1452df82cafa068bdf1a5acd0022d95b13","abstract_canon_sha256":"9f6d10b2579b84ed7a507ca0a8128ffa3d91c87fe3746084e516befc5dcfa1e6"},"schema_version":"1.0"},"canonical_sha256":"f70b8b34896a0fd43c885c361d6fc9dd60ee774832c1336785005d2ba5ef51fe","source":{"kind":"arxiv","id":"2605.14927","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14927","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14927v1","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14927","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"pith_short_12","alias_value":"64FYWNEJNIH5","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"64FYWNEJNIH5IPEI","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"64FYWNEJ","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:64FYWNEJNIH5IPEILQ3B236J3V","target":"record","payload":{"canonical_record":{"source":{"id":"2605.14927","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-14T15:02:24Z","cross_cats_sorted":[],"title_canon_sha256":"2021f781bd8f345eb8a7ac62f5d6ce1452df82cafa068bdf1a5acd0022d95b13","abstract_canon_sha256":"9f6d10b2579b84ed7a507ca0a8128ffa3d91c87fe3746084e516befc5dcfa1e6"},"schema_version":"1.0"},"canonical_sha256":"f70b8b34896a0fd43c885c361d6fc9dd60ee774832c1336785005d2ba5ef51fe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:38:55.603957Z","signature_b64":"Y2+6bqayuYI0KYBEMfKKKR1ycjAxYMMKmjlHwzjjco09PhPFco1rABsyc9oGuUyTOt9RczE43EJ7xTyctucuBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f70b8b34896a0fd43c885c361d6fc9dd60ee774832c1336785005d2ba5ef51fe","last_reissued_at":"2026-05-17T23:38:55.603275Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:38:55.603275Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.14927","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:38:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VaGqJLbjlcofvhvsMkpATIv111dQ9jkX76cFQIZxd8hcZTxWie+2Nw/wMXzlLUfC6/dhqmKHFPEDIxaQiCJnDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T07:16:58.900461Z"},"content_sha256":"46118f5d6ae69cc827e28375a376acf4b5775563854f23954df12123af863717","schema_version":"1.0","event_id":"sha256:46118f5d6ae69cc827e28375a376acf4b5775563854f23954df12123af863717"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:64FYWNEJNIH5IPEILQ3B236J3V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Learning with Shallow Neural Networks on Cluster-Structured Features","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Elisabetta Cornacchia, Laurent Massouli\\'e","submitted_at":"2026-05-14T15:02:24Z","abstract_excerpt":"The success of deep learning in high-dimensional settings is often attributed to the presence of low-dimensional structure in real-world data. While standard theoretical models typically assume that this structure lies in the target function, projecting unstructured inputs onto a low-dimensional subspace, data such as images, text or genomic sequences exhibit strong spatial correlations within the input space itself. In this paper, we propose a tractable model to study how these correlations affect the sample complexity of learning with gradient descent on shallow neural networks. Specifically"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.14927","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:38:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B3AUrQ+xsblwtw7CG3/ORRJXQnzkoNcPrY/VeiEYRphzphnr8YhnAAuP1wMXG6hHJTBG0j+NAPjXYZUzgTs/Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T07:16:58.900796Z"},"content_sha256":"63f122f2e067332156655b4a8b2c9c67e084637833ccdd14d4791a92b62468b0","schema_version":"1.0","event_id":"sha256:63f122f2e067332156655b4a8b2c9c67e084637833ccdd14d4791a92b62468b0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/64FYWNEJNIH5IPEILQ3B236J3V/bundle.json","state_url":"https://pith.science/pith/64FYWNEJNIH5IPEILQ3B236J3V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/64FYWNEJNIH5IPEILQ3B236J3V/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":"2026-06-28T07:16:58Z","links":{"resolver":"https://pith.science/pith/64FYWNEJNIH5IPEILQ3B236J3V","bundle":"https://pith.science/pith/64FYWNEJNIH5IPEILQ3B236J3V/bundle.json","state":"https://pith.science/pith/64FYWNEJNIH5IPEILQ3B236J3V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/64FYWNEJNIH5IPEILQ3B236J3V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:64FYWNEJNIH5IPEILQ3B236J3V","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":"9f6d10b2579b84ed7a507ca0a8128ffa3d91c87fe3746084e516befc5dcfa1e6","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-14T15:02:24Z","title_canon_sha256":"2021f781bd8f345eb8a7ac62f5d6ce1452df82cafa068bdf1a5acd0022d95b13"},"schema_version":"1.0","source":{"id":"2605.14927","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14927","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14927v1","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14927","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"pith_short_12","alias_value":"64FYWNEJNIH5","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"64FYWNEJNIH5IPEI","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"64FYWNEJ","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:63f122f2e067332156655b4a8b2c9c67e084637833ccdd14d4791a92b62468b0","target":"graph","created_at":"2026-05-17T23:38:55Z","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"},"paper":{"abstract_excerpt":"The success of deep learning in high-dimensional settings is often attributed to the presence of low-dimensional structure in real-world data. While standard theoretical models typically assume that this structure lies in the target function, projecting unstructured inputs onto a low-dimensional subspace, data such as images, text or genomic sequences exhibit strong spatial correlations within the input space itself. In this paper, we propose a tractable model to study how these correlations affect the sample complexity of learning with gradient descent on shallow neural networks. Specifically","authors_text":"Elisabetta Cornacchia, Laurent Massouli\\'e","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-14T15:02:24Z","title":"Learning with Shallow Neural Networks on Cluster-Structured Features"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.14927","kind":"arxiv","version":1},"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":"sha256:46118f5d6ae69cc827e28375a376acf4b5775563854f23954df12123af863717","target":"record","created_at":"2026-05-17T23:38:55Z","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":"9f6d10b2579b84ed7a507ca0a8128ffa3d91c87fe3746084e516befc5dcfa1e6","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-05-14T15:02:24Z","title_canon_sha256":"2021f781bd8f345eb8a7ac62f5d6ce1452df82cafa068bdf1a5acd0022d95b13"},"schema_version":"1.0","source":{"id":"2605.14927","kind":"arxiv","version":1}},"canonical_sha256":"f70b8b34896a0fd43c885c361d6fc9dd60ee774832c1336785005d2ba5ef51fe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f70b8b34896a0fd43c885c361d6fc9dd60ee774832c1336785005d2ba5ef51fe","first_computed_at":"2026-05-17T23:38:55.603275Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:38:55.603275Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Y2+6bqayuYI0KYBEMfKKKR1ycjAxYMMKmjlHwzjjco09PhPFco1rABsyc9oGuUyTOt9RczE43EJ7xTyctucuBQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:38:55.603957Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.14927","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:46118f5d6ae69cc827e28375a376acf4b5775563854f23954df12123af863717","sha256:63f122f2e067332156655b4a8b2c9c67e084637833ccdd14d4791a92b62468b0"],"state_sha256":"135f839113a4cb0ce2492f906e16618d51818a07d8c8e30a07eb88b56d3dd948"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nj5tAQKXV2UMrHqN3ICNAubSY/3Y68tPwFIlBKna+GevSrk/Ph5WcLunl2CqhCp3kGpjE/5Ql+l1JfJ9fQZPDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T07:16:58.902631Z","bundle_sha256":"892fc36436bb5951fc50bdc5dc2a08005ea1fa45dbcb64a5739b0ef048270111"}}