{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:UX2LMDNY3NDKCVYYTFNVXONODI","short_pith_number":"pith:UX2LMDNY","canonical_record":{"source":{"id":"2306.12308","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2023-06-21T14:39:44Z","cross_cats_sorted":["cs.IT","math.IT","stat.TH"],"title_canon_sha256":"687bcd9083728943d73bba0bb6ef29131715c7a82c46968ddef24639422e2ba7","abstract_canon_sha256":"e941e6de5d5cefcff42846dda5cbf25de440b38f233676f5421191d6bb4d4c08"},"schema_version":"1.0"},"canonical_sha256":"a5f4b60db8db46a15718995b5bb9ae1a2a34005c54c411d661d0c1ae0908bebe","source":{"kind":"arxiv","id":"2306.12308","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2306.12308","created_at":"2026-07-05T06:24:50Z"},{"alias_kind":"arxiv_version","alias_value":"2306.12308v2","created_at":"2026-07-05T06:24:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2306.12308","created_at":"2026-07-05T06:24:50Z"},{"alias_kind":"pith_short_12","alias_value":"UX2LMDNY3NDK","created_at":"2026-07-05T06:24:50Z"},{"alias_kind":"pith_short_16","alias_value":"UX2LMDNY3NDKCVYY","created_at":"2026-07-05T06:24:50Z"},{"alias_kind":"pith_short_8","alias_value":"UX2LMDNY","created_at":"2026-07-05T06:24:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:UX2LMDNY3NDKCVYYTFNVXONODI","target":"record","payload":{"canonical_record":{"source":{"id":"2306.12308","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2023-06-21T14:39:44Z","cross_cats_sorted":["cs.IT","math.IT","stat.TH"],"title_canon_sha256":"687bcd9083728943d73bba0bb6ef29131715c7a82c46968ddef24639422e2ba7","abstract_canon_sha256":"e941e6de5d5cefcff42846dda5cbf25de440b38f233676f5421191d6bb4d4c08"},"schema_version":"1.0"},"canonical_sha256":"a5f4b60db8db46a15718995b5bb9ae1a2a34005c54c411d661d0c1ae0908bebe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:24:50.922831Z","signature_b64":"aJSwBL4/KRerUozc5NtG8N3oLYuLuvUmnI/Q+9b4zR1Mr+v5KXw0rvK98ds9Xm/VbdSt08xeAPGAh0K5MAk3Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5f4b60db8db46a15718995b5bb9ae1a2a34005c54c411d661d0c1ae0908bebe","last_reissued_at":"2026-07-05T06:24:50.922403Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:24:50.922403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2306.12308","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":"2026-07-05T06:24:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"duDavC4MW0Ja63v8A8uW751080qgblvQRHB3J+U+nwIKui+RZ7gcjXajOGnu00R35xEzhZom9bIq0isFUEpOBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T04:14:01.062672Z"},"content_sha256":"642f30ca2f69a8053bdea76c1b6ef980ed5a216d35191ceede0ee7149fb8661a","schema_version":"1.0","event_id":"sha256:642f30ca2f69a8053bdea76c1b6ef980ed5a216d35191ceede0ee7149fb8661a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:UX2LMDNY3NDKCVYYTFNVXONODI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Entropic characterization of optimal rates for learning Gaussian mixtures","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.IT","math.IT","stat.TH"],"primary_cat":"math.ST","authors_text":"Yihong Wu, Yury Polyanskiy, Zeyu Jia","submitted_at":"2023-06-21T14:39:44Z","abstract_excerpt":"We consider the question of estimating multi-dimensional Gaussian mixtures (GM) with compactly supported or subgaussian mixing distributions. Minimax estimation rate for this class (under Hellinger, TV and KL divergences) is a long-standing open question, even in one dimension. In this paper we characterize this rate (for all constant dimensions) in terms of the metric entropy of the class. Such characterizations originate from seminal works of Le Cam (1973); Birge (1983); Haussler and Opper (1997); Yang and Barron (1999). However, for GMs a key ingredient missing from earlier work (and widely"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.12308","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/2306.12308/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":"2026-07-05T06:24:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"duyfeEelyB8H+6eMwm4APhKlZTIZSNsn+jdiFcv4krMvb2Us7J6aiMqCqhdczWOuQCvVTuycmb7IGbJp9D3fCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T04:14:01.063044Z"},"content_sha256":"a58096a3b8b79834c219e60b4bf911da593eea1ffc82770755ca086c921c85b6","schema_version":"1.0","event_id":"sha256:a58096a3b8b79834c219e60b4bf911da593eea1ffc82770755ca086c921c85b6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UX2LMDNY3NDKCVYYTFNVXONODI/bundle.json","state_url":"https://pith.science/pith/UX2LMDNY3NDKCVYYTFNVXONODI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UX2LMDNY3NDKCVYYTFNVXONODI/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":"2026-07-07T04:14:01Z","links":{"resolver":"https://pith.science/pith/UX2LMDNY3NDKCVYYTFNVXONODI","bundle":"https://pith.science/pith/UX2LMDNY3NDKCVYYTFNVXONODI/bundle.json","state":"https://pith.science/pith/UX2LMDNY3NDKCVYYTFNVXONODI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UX2LMDNY3NDKCVYYTFNVXONODI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:UX2LMDNY3NDKCVYYTFNVXONODI","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":"e941e6de5d5cefcff42846dda5cbf25de440b38f233676f5421191d6bb4d4c08","cross_cats_sorted":["cs.IT","math.IT","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2023-06-21T14:39:44Z","title_canon_sha256":"687bcd9083728943d73bba0bb6ef29131715c7a82c46968ddef24639422e2ba7"},"schema_version":"1.0","source":{"id":"2306.12308","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2306.12308","created_at":"2026-07-05T06:24:50Z"},{"alias_kind":"arxiv_version","alias_value":"2306.12308v2","created_at":"2026-07-05T06:24:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2306.12308","created_at":"2026-07-05T06:24:50Z"},{"alias_kind":"pith_short_12","alias_value":"UX2LMDNY3NDK","created_at":"2026-07-05T06:24:50Z"},{"alias_kind":"pith_short_16","alias_value":"UX2LMDNY3NDKCVYY","created_at":"2026-07-05T06:24:50Z"},{"alias_kind":"pith_short_8","alias_value":"UX2LMDNY","created_at":"2026-07-05T06:24:50Z"}],"graph_snapshots":[{"event_id":"sha256:a58096a3b8b79834c219e60b4bf911da593eea1ffc82770755ca086c921c85b6","target":"graph","created_at":"2026-07-05T06:24:50Z","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/2306.12308/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider the question of estimating multi-dimensional Gaussian mixtures (GM) with compactly supported or subgaussian mixing distributions. Minimax estimation rate for this class (under Hellinger, TV and KL divergences) is a long-standing open question, even in one dimension. In this paper we characterize this rate (for all constant dimensions) in terms of the metric entropy of the class. Such characterizations originate from seminal works of Le Cam (1973); Birge (1983); Haussler and Opper (1997); Yang and Barron (1999). However, for GMs a key ingredient missing from earlier work (and widely","authors_text":"Yihong Wu, Yury Polyanskiy, Zeyu Jia","cross_cats":["cs.IT","math.IT","stat.TH"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2023-06-21T14:39:44Z","title":"Entropic characterization of optimal rates for learning Gaussian mixtures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.12308","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":"sha256:642f30ca2f69a8053bdea76c1b6ef980ed5a216d35191ceede0ee7149fb8661a","target":"record","created_at":"2026-07-05T06:24:50Z","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":"e941e6de5d5cefcff42846dda5cbf25de440b38f233676f5421191d6bb4d4c08","cross_cats_sorted":["cs.IT","math.IT","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.ST","submitted_at":"2023-06-21T14:39:44Z","title_canon_sha256":"687bcd9083728943d73bba0bb6ef29131715c7a82c46968ddef24639422e2ba7"},"schema_version":"1.0","source":{"id":"2306.12308","kind":"arxiv","version":2}},"canonical_sha256":"a5f4b60db8db46a15718995b5bb9ae1a2a34005c54c411d661d0c1ae0908bebe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a5f4b60db8db46a15718995b5bb9ae1a2a34005c54c411d661d0c1ae0908bebe","first_computed_at":"2026-07-05T06:24:50.922403Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:24:50.922403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aJSwBL4/KRerUozc5NtG8N3oLYuLuvUmnI/Q+9b4zR1Mr+v5KXw0rvK98ds9Xm/VbdSt08xeAPGAh0K5MAk3Dw==","signature_status":"signed_v1","signed_at":"2026-07-05T06:24:50.922831Z","signed_message":"canonical_sha256_bytes"},"source_id":"2306.12308","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:642f30ca2f69a8053bdea76c1b6ef980ed5a216d35191ceede0ee7149fb8661a","sha256:a58096a3b8b79834c219e60b4bf911da593eea1ffc82770755ca086c921c85b6"],"state_sha256":"78bd29a6adda264995f8b8f4f154f97571bbaba943e337030443f5e8413b92d4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Bx7GVOdoyZSo/52owqLw9F6gvKQ7SXmt3QEY5+EVpXX37T/oFKtZjHVFMQHrmw3BVFB4sq1cAKKOgVIuGtMHCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T04:14:01.065092Z","bundle_sha256":"b4ffa29a0dd1572c629123ba0b76f38a4747d64b7d7bee504565fbf07e3977e5"}}