{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:MLUWJFUBYGFVBUZ6NP5NPC2KT4","short_pith_number":"pith:MLUWJFUB","canonical_record":{"source":{"id":"2505.20433","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2025-05-26T18:27:17Z","cross_cats_sorted":["cs.LG","math.ST","stat.TH"],"title_canon_sha256":"b36517c95851ac3207fc0b7b2871bb4b12ac0fce74c68f1244af7e443e171929","abstract_canon_sha256":"6d8a1118b38370958593b47de362d2f6a2e6855f13e717cbe4fcef9057ff14fe"},"schema_version":"1.0"},"canonical_sha256":"62e9649681c18b50d33e6bfad78b4a9f3bfaead77dca840649a204fc1a4216e6","source":{"kind":"arxiv","id":"2505.20433","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.20433","created_at":"2026-07-05T11:10:12Z"},{"alias_kind":"arxiv_version","alias_value":"2505.20433v1","created_at":"2026-07-05T11:10:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.20433","created_at":"2026-07-05T11:10:12Z"},{"alias_kind":"pith_short_12","alias_value":"MLUWJFUBYGFV","created_at":"2026-07-05T11:10:12Z"},{"alias_kind":"pith_short_16","alias_value":"MLUWJFUBYGFVBUZ6","created_at":"2026-07-05T11:10:12Z"},{"alias_kind":"pith_short_8","alias_value":"MLUWJFUB","created_at":"2026-07-05T11:10:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:MLUWJFUBYGFVBUZ6NP5NPC2KT4","target":"record","payload":{"canonical_record":{"source":{"id":"2505.20433","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2025-05-26T18:27:17Z","cross_cats_sorted":["cs.LG","math.ST","stat.TH"],"title_canon_sha256":"b36517c95851ac3207fc0b7b2871bb4b12ac0fce74c68f1244af7e443e171929","abstract_canon_sha256":"6d8a1118b38370958593b47de362d2f6a2e6855f13e717cbe4fcef9057ff14fe"},"schema_version":"1.0"},"canonical_sha256":"62e9649681c18b50d33e6bfad78b4a9f3bfaead77dca840649a204fc1a4216e6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:10:12.438915Z","signature_b64":"jyGBvQWTkXw0ryGYPiIMRhEf+KJrUzn2a3zHYeCjsEXRVM7EHohoAjBSIjMBBZH0pPprL2xLZ6KzDET5S4Y5DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62e9649681c18b50d33e6bfad78b4a9f3bfaead77dca840649a204fc1a4216e6","last_reissued_at":"2026-07-05T11:10:12.438435Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:10:12.438435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2505.20433","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":"2026-07-05T11:10:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1j7J33TjCn7je3Rc4LRnmaBFdog1mnoHZC0SfSR931Ud+3RpNn33XON6Fp5hNIeQOQx+cDF56X9RAKQzLdQjBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T05:53:20.380235Z"},"content_sha256":"b65a61a4fb66908efb12afa63d5d8690f1d74350b2d146a7c87d085feb93c3b4","schema_version":"1.0","event_id":"sha256:b65a61a4fb66908efb12afa63d5d8690f1d74350b2d146a7c87d085feb93c3b4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:MLUWJFUBYGFVBUZ6NP5NPC2KT4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kernel Quantile Embeddings and Associated Probability Metrics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Fran\\c{c}ois-Xavier Briol, Krikamol Muandet, Masha Naslidnyk, Siu Lun Chau","submitted_at":"2025-05-26T18:27:17Z","abstract_excerpt":"Embedding probability distributions into reproducing kernel Hilbert spaces (RKHS) has enabled powerful nonparametric methods such as the maximum mean discrepancy (MMD), a statistical distance with strong theoretical and computational properties. At its core, the MMD relies on kernel mean embeddings to represent distributions as mean functions in RKHS. However, it remains unclear if the mean function is the only meaningful RKHS representation. Inspired by generalised quantiles, we introduce the notion of kernel quantile embeddings (KQEs). We then use KQEs to construct a family of distances that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.20433","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/2505.20433/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":"2026-07-05T11:10:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GUSL7ScWZIlgWQygHZ/fCHPD/0SKmG44M6l2FtS8oX7VAUdpK+sVbY51rIvKcoS81nxKgYE/1sGa2WpO+WnsDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T05:53:20.380616Z"},"content_sha256":"9e71edf01630992807870e842f7aa379bd21cb9f3c3dc08c8ca020fcbaa1a3e7","schema_version":"1.0","event_id":"sha256:9e71edf01630992807870e842f7aa379bd21cb9f3c3dc08c8ca020fcbaa1a3e7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MLUWJFUBYGFVBUZ6NP5NPC2KT4/bundle.json","state_url":"https://pith.science/pith/MLUWJFUBYGFVBUZ6NP5NPC2KT4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MLUWJFUBYGFVBUZ6NP5NPC2KT4/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":"2026-08-22T05:53:20Z","links":{"resolver":"https://pith.science/pith/MLUWJFUBYGFVBUZ6NP5NPC2KT4","bundle":"https://pith.science/pith/MLUWJFUBYGFVBUZ6NP5NPC2KT4/bundle.json","state":"https://pith.science/pith/MLUWJFUBYGFVBUZ6NP5NPC2KT4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MLUWJFUBYGFVBUZ6NP5NPC2KT4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:MLUWJFUBYGFVBUZ6NP5NPC2KT4","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":"6d8a1118b38370958593b47de362d2f6a2e6855f13e717cbe4fcef9057ff14fe","cross_cats_sorted":["cs.LG","math.ST","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2025-05-26T18:27:17Z","title_canon_sha256":"b36517c95851ac3207fc0b7b2871bb4b12ac0fce74c68f1244af7e443e171929"},"schema_version":"1.0","source":{"id":"2505.20433","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.20433","created_at":"2026-07-05T11:10:12Z"},{"alias_kind":"arxiv_version","alias_value":"2505.20433v1","created_at":"2026-07-05T11:10:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.20433","created_at":"2026-07-05T11:10:12Z"},{"alias_kind":"pith_short_12","alias_value":"MLUWJFUBYGFV","created_at":"2026-07-05T11:10:12Z"},{"alias_kind":"pith_short_16","alias_value":"MLUWJFUBYGFVBUZ6","created_at":"2026-07-05T11:10:12Z"},{"alias_kind":"pith_short_8","alias_value":"MLUWJFUB","created_at":"2026-07-05T11:10:12Z"}],"graph_snapshots":[{"event_id":"sha256:9e71edf01630992807870e842f7aa379bd21cb9f3c3dc08c8ca020fcbaa1a3e7","target":"graph","created_at":"2026-07-05T11:10:12Z","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/2505.20433/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Embedding probability distributions into reproducing kernel Hilbert spaces (RKHS) has enabled powerful nonparametric methods such as the maximum mean discrepancy (MMD), a statistical distance with strong theoretical and computational properties. At its core, the MMD relies on kernel mean embeddings to represent distributions as mean functions in RKHS. However, it remains unclear if the mean function is the only meaningful RKHS representation. Inspired by generalised quantiles, we introduce the notion of kernel quantile embeddings (KQEs). We then use KQEs to construct a family of distances that","authors_text":"Fran\\c{c}ois-Xavier Briol, Krikamol Muandet, Masha Naslidnyk, Siu Lun Chau","cross_cats":["cs.LG","math.ST","stat.TH"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2025-05-26T18:27:17Z","title":"Kernel Quantile Embeddings and Associated Probability Metrics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.20433","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:b65a61a4fb66908efb12afa63d5d8690f1d74350b2d146a7c87d085feb93c3b4","target":"record","created_at":"2026-07-05T11:10:12Z","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":"6d8a1118b38370958593b47de362d2f6a2e6855f13e717cbe4fcef9057ff14fe","cross_cats_sorted":["cs.LG","math.ST","stat.TH"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"stat.ML","submitted_at":"2025-05-26T18:27:17Z","title_canon_sha256":"b36517c95851ac3207fc0b7b2871bb4b12ac0fce74c68f1244af7e443e171929"},"schema_version":"1.0","source":{"id":"2505.20433","kind":"arxiv","version":1}},"canonical_sha256":"62e9649681c18b50d33e6bfad78b4a9f3bfaead77dca840649a204fc1a4216e6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62e9649681c18b50d33e6bfad78b4a9f3bfaead77dca840649a204fc1a4216e6","first_computed_at":"2026-07-05T11:10:12.438435Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:10:12.438435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jyGBvQWTkXw0ryGYPiIMRhEf+KJrUzn2a3zHYeCjsEXRVM7EHohoAjBSIjMBBZH0pPprL2xLZ6KzDET5S4Y5DA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:10:12.438915Z","signed_message":"canonical_sha256_bytes"},"source_id":"2505.20433","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b65a61a4fb66908efb12afa63d5d8690f1d74350b2d146a7c87d085feb93c3b4","sha256:9e71edf01630992807870e842f7aa379bd21cb9f3c3dc08c8ca020fcbaa1a3e7"],"state_sha256":"7b5dd4932bc824e0801bd362308bfa6a1b042cbfa670f8b84e26c2fd63877790"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5IQFJoUw+N04mjY03vtVH7DDYJUefAu0Uh190yhBoV+OanrCjVOdtr3uabcTP3AZWdL9EdlEB2Xr4XpH+9jLBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T05:53:20.383517Z","bundle_sha256":"bcf90308965197bd420f66610768a378f492af7e86342f553cb42c2245682498"}}