{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:LIISJUUCHJLBRFHAADFJ6Y67WD","short_pith_number":"pith:LIISJUUC","canonical_record":{"source":{"id":"2402.00423","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2024-02-01T08:44:16Z","cross_cats_sorted":["math.PR","stat.TH"],"title_canon_sha256":"389464aa747de3d3b6212a5464ae1e48836d94db3caa5a2a9ff95a5145439816","abstract_canon_sha256":"b871de549eedd9021e166c451b5414dbe73e27fd7bc464fce76c182d610970ea"},"schema_version":"1.0"},"canonical_sha256":"5a1124d2823a561894e000ca9f63dfb0fc90de28aeaed6efe0977798fd0271ae","source":{"kind":"arxiv","id":"2402.00423","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.00423","created_at":"2026-07-05T08:22:36Z"},{"alias_kind":"arxiv_version","alias_value":"2402.00423v2","created_at":"2026-07-05T08:22:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.00423","created_at":"2026-07-05T08:22:36Z"},{"alias_kind":"pith_short_12","alias_value":"LIISJUUCHJLB","created_at":"2026-07-05T08:22:36Z"},{"alias_kind":"pith_short_16","alias_value":"LIISJUUCHJLBRFHA","created_at":"2026-07-05T08:22:36Z"},{"alias_kind":"pith_short_8","alias_value":"LIISJUUC","created_at":"2026-07-05T08:22:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:LIISJUUCHJLBRFHAADFJ6Y67WD","target":"record","payload":{"canonical_record":{"source":{"id":"2402.00423","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2024-02-01T08:44:16Z","cross_cats_sorted":["math.PR","stat.TH"],"title_canon_sha256":"389464aa747de3d3b6212a5464ae1e48836d94db3caa5a2a9ff95a5145439816","abstract_canon_sha256":"b871de549eedd9021e166c451b5414dbe73e27fd7bc464fce76c182d610970ea"},"schema_version":"1.0"},"canonical_sha256":"5a1124d2823a561894e000ca9f63dfb0fc90de28aeaed6efe0977798fd0271ae","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:22:36.577189Z","signature_b64":"5a/aFBZ9CTVxOuUzCZTfj00P4rULZ9xjAd16Ng4ds82M8ST1vmRP3mr7tPu+if61Ix8F/L6W7yUPANVe2b0TAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5a1124d2823a561894e000ca9f63dfb0fc90de28aeaed6efe0977798fd0271ae","last_reissued_at":"2026-07-05T08:22:36.576634Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:22:36.576634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2402.00423","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":"2026-07-05T08:22:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HkQHuZ2FstDJrcYSyB5ooxCrt9/XH6fpkXsAxSrMl6Iz2XlfIDhoEPReX1Ol0gAGFaJES9sgM3rSGD4UBEbABA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-10T06:11:37.552943Z"},"content_sha256":"f7df38fe0f25c02812eb8a9a7cc3d2ab9f36e8520d1e9e1df60f31761e0734b4","schema_version":"1.0","event_id":"sha256:f7df38fe0f25c02812eb8a9a7cc3d2ab9f36e8520d1e9e1df60f31761e0734b4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:LIISJUUCHJLBRFHAADFJ6Y67WD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hierarchical Integral Probability Metrics: A distance on random probability measures with low sample complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR","stat.TH"],"primary_cat":"math.ST","authors_text":"Hugo Lavenant, Marta Catalano","submitted_at":"2024-02-01T08:44:16Z","abstract_excerpt":"Random probabilities are a key component to many nonparametric methods in Statistics and Machine Learning. To quantify comparisons between different laws of random probabilities several works are starting to use the elegant Wasserstein over Wasserstein distance. In this paper we prove that the infinite dimensionality of the space of probabilities drastically deteriorates its sample complexity, which is slower than any polynomial rate in the sample size. We propose a new distance that preserves many desirable properties of the former while achieving a parametric rate of convergence. In particul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.00423","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/2402.00423/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":"2026-07-05T08:22:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eX7XjF10RMrdB+ObqEUyG1q31F3D6Kjso3ITy4RtL7ZXMCjnNvtqZRv7W6XOXLm0lnl0zN3p8V1r5z1i8/DJCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-10T06:11:37.553331Z"},"content_sha256":"04a1bca899b42b7a7a909d9506afddfbe94dabd199b59c9d8c2fe21617ec5f87","schema_version":"1.0","event_id":"sha256:04a1bca899b42b7a7a909d9506afddfbe94dabd199b59c9d8c2fe21617ec5f87"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LIISJUUCHJLBRFHAADFJ6Y67WD/bundle.json","state_url":"https://pith.science/pith/LIISJUUCHJLBRFHAADFJ6Y67WD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LIISJUUCHJLBRFHAADFJ6Y67WD/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":"2026-07-10T06:11:37Z","links":{"resolver":"https://pith.science/pith/LIISJUUCHJLBRFHAADFJ6Y67WD","bundle":"https://pith.science/pith/LIISJUUCHJLBRFHAADFJ6Y67WD/bundle.json","state":"https://pith.science/pith/LIISJUUCHJLBRFHAADFJ6Y67WD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LIISJUUCHJLBRFHAADFJ6Y67WD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:LIISJUUCHJLBRFHAADFJ6Y67WD","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":"b871de549eedd9021e166c451b5414dbe73e27fd7bc464fce76c182d610970ea","cross_cats_sorted":["math.PR","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2024-02-01T08:44:16Z","title_canon_sha256":"389464aa747de3d3b6212a5464ae1e48836d94db3caa5a2a9ff95a5145439816"},"schema_version":"1.0","source":{"id":"2402.00423","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.00423","created_at":"2026-07-05T08:22:36Z"},{"alias_kind":"arxiv_version","alias_value":"2402.00423v2","created_at":"2026-07-05T08:22:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.00423","created_at":"2026-07-05T08:22:36Z"},{"alias_kind":"pith_short_12","alias_value":"LIISJUUCHJLB","created_at":"2026-07-05T08:22:36Z"},{"alias_kind":"pith_short_16","alias_value":"LIISJUUCHJLBRFHA","created_at":"2026-07-05T08:22:36Z"},{"alias_kind":"pith_short_8","alias_value":"LIISJUUC","created_at":"2026-07-05T08:22:36Z"}],"graph_snapshots":[{"event_id":"sha256:04a1bca899b42b7a7a909d9506afddfbe94dabd199b59c9d8c2fe21617ec5f87","target":"graph","created_at":"2026-07-05T08:22:36Z","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/2402.00423/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Random probabilities are a key component to many nonparametric methods in Statistics and Machine Learning. To quantify comparisons between different laws of random probabilities several works are starting to use the elegant Wasserstein over Wasserstein distance. In this paper we prove that the infinite dimensionality of the space of probabilities drastically deteriorates its sample complexity, which is slower than any polynomial rate in the sample size. We propose a new distance that preserves many desirable properties of the former while achieving a parametric rate of convergence. In particul","authors_text":"Hugo Lavenant, Marta Catalano","cross_cats":["math.PR","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2024-02-01T08:44:16Z","title":"Hierarchical Integral Probability Metrics: A distance on random probability measures with low sample complexity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.00423","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:f7df38fe0f25c02812eb8a9a7cc3d2ab9f36e8520d1e9e1df60f31761e0734b4","target":"record","created_at":"2026-07-05T08:22:36Z","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":"b871de549eedd9021e166c451b5414dbe73e27fd7bc464fce76c182d610970ea","cross_cats_sorted":["math.PR","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2024-02-01T08:44:16Z","title_canon_sha256":"389464aa747de3d3b6212a5464ae1e48836d94db3caa5a2a9ff95a5145439816"},"schema_version":"1.0","source":{"id":"2402.00423","kind":"arxiv","version":2}},"canonical_sha256":"5a1124d2823a561894e000ca9f63dfb0fc90de28aeaed6efe0977798fd0271ae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5a1124d2823a561894e000ca9f63dfb0fc90de28aeaed6efe0977798fd0271ae","first_computed_at":"2026-07-05T08:22:36.576634Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:22:36.576634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5a/aFBZ9CTVxOuUzCZTfj00P4rULZ9xjAd16Ng4ds82M8ST1vmRP3mr7tPu+if61Ix8F/L6W7yUPANVe2b0TAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T08:22:36.577189Z","signed_message":"canonical_sha256_bytes"},"source_id":"2402.00423","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f7df38fe0f25c02812eb8a9a7cc3d2ab9f36e8520d1e9e1df60f31761e0734b4","sha256:04a1bca899b42b7a7a909d9506afddfbe94dabd199b59c9d8c2fe21617ec5f87"],"state_sha256":"f61f6e39c9955090d57f2300d7856e208b926ec1e899e3b7eef346806858f5c4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8hCY30IkdZaE+ytSSzERN0wnwahO7ok8PzhMkckw01sjMdjsABfnXIjLM2aOOAcGqtUcfaJHBc1CzayaSmmVCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-10T06:11:37.555491Z","bundle_sha256":"cb23fc5fbd04cf418d5675cab3dce93910bbdb6cecad73a32860efd94a1d829b"}}