{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:AWA64TUEJEOK5L2JNBH7HPCYQG","short_pith_number":"pith:AWA64TUE","canonical_record":{"source":{"id":"2509.10355","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-09-12T15:43:13Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"9d63bb7b975fcee51cf88423592c8b282cd0567dd8afacac8c57fc3c29b4a3cc","abstract_canon_sha256":"95d32b3a2cb3e5e8a103de80d1def03440b467291078de429a549453bd98dbd6"},"schema_version":"1.0"},"canonical_sha256":"0581ee4e84491caeaf49684ff3bc58818286673e2d463a28e2c433592176e3db","source":{"kind":"arxiv","id":"2509.10355","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.10355","created_at":"2026-07-05T12:11:08Z"},{"alias_kind":"arxiv_version","alias_value":"2509.10355v1","created_at":"2026-07-05T12:11:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.10355","created_at":"2026-07-05T12:11:08Z"},{"alias_kind":"pith_short_12","alias_value":"AWA64TUEJEOK","created_at":"2026-07-05T12:11:08Z"},{"alias_kind":"pith_short_16","alias_value":"AWA64TUEJEOK5L2J","created_at":"2026-07-05T12:11:08Z"},{"alias_kind":"pith_short_8","alias_value":"AWA64TUE","created_at":"2026-07-05T12:11:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:AWA64TUEJEOK5L2JNBH7HPCYQG","target":"record","payload":{"canonical_record":{"source":{"id":"2509.10355","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-09-12T15:43:13Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"9d63bb7b975fcee51cf88423592c8b282cd0567dd8afacac8c57fc3c29b4a3cc","abstract_canon_sha256":"95d32b3a2cb3e5e8a103de80d1def03440b467291078de429a549453bd98dbd6"},"schema_version":"1.0"},"canonical_sha256":"0581ee4e84491caeaf49684ff3bc58818286673e2d463a28e2c433592176e3db","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:11:08.306087Z","signature_b64":"K+4MfCwjImTzKsASDW53zUDAmK0jVzxwnY6nCHDcU4MCbGM+ASo+dyEQaW4o73zjaDQh9H8Gzeuz2b26l1FSBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0581ee4e84491caeaf49684ff3bc58818286673e2d463a28e2c433592176e3db","last_reissued_at":"2026-07-05T12:11:08.305500Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:11:08.305500Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.10355","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":"2026-07-05T12:11:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U9Iw0st4rgPwSpka7/dfnhaPyKIc6UZ3haPLyKHqbKNRc8Chy0+Y7s5QZt7C2w8zZv1iWidzq2lE8HKJC3cDDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T06:20:44.072600Z"},"content_sha256":"361402c8195a3c9976e42d902a6a934c8b3491ec322947e3ee750292a856843d","schema_version":"1.0","event_id":"sha256:361402c8195a3c9976e42d902a6a934c8b3491ec322947e3ee750292a856843d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:AWA64TUEJEOK5L2JNBH7HPCYQG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Entropy and Learning of Lipschitz Functions under Log-Concave Measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Boaz Klartag, Pierre Bizeul","submitted_at":"2025-09-12T15:43:13Z","abstract_excerpt":"We study regression of $1$-Lipschitz functions under a log-concave measure $\\mu$ on $\\mathbb{R}^d$. We focus on the high-dimensional regime where the sample size $n$ is subexponential in $d$, in which distribution-free estimators are ineffective. We analyze two polynomial-based procedures: the projection estimator, which relies on knowledge of an orthogonal polynomial basis of $\\mu$, and the least-squares estimator over low-degree polynomials, which requires no knowledge of $\\mu$ whatsoever. Their risk is governed by the rate of polynomial approximation of Lipschitz functions in $L^2(\\mu)$. Wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.10355","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.10355/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":"2026-07-05T12:11:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I6fxBANzxog5nn3N8NFrSpFDR6fFo0s+LS25NIqV5f194VJX4IINR5qehbYo+vkVxlSwSzQBDMJdsQWqao4PBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T06:20:44.073625Z"},"content_sha256":"2640de848d59a49e6a1b5b326115fe58a53c35b704b6c5743667f4c8822e7052","schema_version":"1.0","event_id":"sha256:2640de848d59a49e6a1b5b326115fe58a53c35b704b6c5743667f4c8822e7052"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AWA64TUEJEOK5L2JNBH7HPCYQG/bundle.json","state_url":"https://pith.science/pith/AWA64TUEJEOK5L2JNBH7HPCYQG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AWA64TUEJEOK5L2JNBH7HPCYQG/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":"2026-08-18T06:20:44Z","links":{"resolver":"https://pith.science/pith/AWA64TUEJEOK5L2JNBH7HPCYQG","bundle":"https://pith.science/pith/AWA64TUEJEOK5L2JNBH7HPCYQG/bundle.json","state":"https://pith.science/pith/AWA64TUEJEOK5L2JNBH7HPCYQG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AWA64TUEJEOK5L2JNBH7HPCYQG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:AWA64TUEJEOK5L2JNBH7HPCYQG","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":"95d32b3a2cb3e5e8a103de80d1def03440b467291078de429a549453bd98dbd6","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-09-12T15:43:13Z","title_canon_sha256":"9d63bb7b975fcee51cf88423592c8b282cd0567dd8afacac8c57fc3c29b4a3cc"},"schema_version":"1.0","source":{"id":"2509.10355","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.10355","created_at":"2026-07-05T12:11:08Z"},{"alias_kind":"arxiv_version","alias_value":"2509.10355v1","created_at":"2026-07-05T12:11:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.10355","created_at":"2026-07-05T12:11:08Z"},{"alias_kind":"pith_short_12","alias_value":"AWA64TUEJEOK","created_at":"2026-07-05T12:11:08Z"},{"alias_kind":"pith_short_16","alias_value":"AWA64TUEJEOK5L2J","created_at":"2026-07-05T12:11:08Z"},{"alias_kind":"pith_short_8","alias_value":"AWA64TUE","created_at":"2026-07-05T12:11:08Z"}],"graph_snapshots":[{"event_id":"sha256:2640de848d59a49e6a1b5b326115fe58a53c35b704b6c5743667f4c8822e7052","target":"graph","created_at":"2026-07-05T12:11:08Z","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/2509.10355/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study regression of $1$-Lipschitz functions under a log-concave measure $\\mu$ on $\\mathbb{R}^d$. We focus on the high-dimensional regime where the sample size $n$ is subexponential in $d$, in which distribution-free estimators are ineffective. We analyze two polynomial-based procedures: the projection estimator, which relies on knowledge of an orthogonal polynomial basis of $\\mu$, and the least-squares estimator over low-degree polynomials, which requires no knowledge of $\\mu$ whatsoever. Their risk is governed by the rate of polynomial approximation of Lipschitz functions in $L^2(\\mu)$. Wh","authors_text":"Boaz Klartag, Pierre Bizeul","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-09-12T15:43:13Z","title":"Entropy and Learning of Lipschitz Functions under Log-Concave Measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.10355","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":"sha256:361402c8195a3c9976e42d902a6a934c8b3491ec322947e3ee750292a856843d","target":"record","created_at":"2026-07-05T12:11:08Z","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":"95d32b3a2cb3e5e8a103de80d1def03440b467291078de429a549453bd98dbd6","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-09-12T15:43:13Z","title_canon_sha256":"9d63bb7b975fcee51cf88423592c8b282cd0567dd8afacac8c57fc3c29b4a3cc"},"schema_version":"1.0","source":{"id":"2509.10355","kind":"arxiv","version":1}},"canonical_sha256":"0581ee4e84491caeaf49684ff3bc58818286673e2d463a28e2c433592176e3db","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0581ee4e84491caeaf49684ff3bc58818286673e2d463a28e2c433592176e3db","first_computed_at":"2026-07-05T12:11:08.305500Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:11:08.305500Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"K+4MfCwjImTzKsASDW53zUDAmK0jVzxwnY6nCHDcU4MCbGM+ASo+dyEQaW4o73zjaDQh9H8Gzeuz2b26l1FSBw==","signature_status":"signed_v1","signed_at":"2026-07-05T12:11:08.306087Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.10355","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:361402c8195a3c9976e42d902a6a934c8b3491ec322947e3ee750292a856843d","sha256:2640de848d59a49e6a1b5b326115fe58a53c35b704b6c5743667f4c8822e7052"],"state_sha256":"84c37b29ca51f565ef51f4368b37934cb04a67880f23455cd47170dc1fab3bbd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vCIM78Vd6Y8c3URpZO47U9N51dDQ/ByCZtJzCCfMwVf5Euw7DlATXRTQTLEjDiLSdcPfn1QuVQLWXsBq2NfvDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T06:20:44.078135Z","bundle_sha256":"0bb6a7b048ec999b388aa19c2aa418638ff4b0ca2669bc04790f89af5a7e14a4"}}