{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:CSRRTJ2ZIZXTML7NTAV23U7EQ4","short_pith_number":"pith:CSRRTJ2Z","canonical_record":{"source":{"id":"2602.02190","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"stat.ML","submitted_at":"2026-02-02T14:56:58Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"63d32ba623176300dfe4ab9ac2e34cac0a2eb2dc86f82f9b62ea95327ac1b1b9","abstract_canon_sha256":"f6ddc5ea32cbb5f23eed8812d2983a8c19153bc0964adfce6b1ded0b3bc3e697"},"schema_version":"1.0"},"canonical_sha256":"14a319a759466f362fed982badd3e48727adcf199d24daf5ec709de4f260363f","source":{"kind":"arxiv","id":"2602.02190","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.02190","created_at":"2026-07-07T02:18:33Z"},{"alias_kind":"arxiv_version","alias_value":"2602.02190v2","created_at":"2026-07-07T02:18:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.02190","created_at":"2026-07-07T02:18:33Z"},{"alias_kind":"pith_short_12","alias_value":"CSRRTJ2ZIZXT","created_at":"2026-07-07T02:18:33Z"},{"alias_kind":"pith_short_16","alias_value":"CSRRTJ2ZIZXTML7N","created_at":"2026-07-07T02:18:33Z"},{"alias_kind":"pith_short_8","alias_value":"CSRRTJ2Z","created_at":"2026-07-07T02:18:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:CSRRTJ2ZIZXTML7NTAV23U7EQ4","target":"record","payload":{"canonical_record":{"source":{"id":"2602.02190","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"stat.ML","submitted_at":"2026-02-02T14:56:58Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"63d32ba623176300dfe4ab9ac2e34cac0a2eb2dc86f82f9b62ea95327ac1b1b9","abstract_canon_sha256":"f6ddc5ea32cbb5f23eed8812d2983a8c19153bc0964adfce6b1ded0b3bc3e697"},"schema_version":"1.0"},"canonical_sha256":"14a319a759466f362fed982badd3e48727adcf199d24daf5ec709de4f260363f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:18:33.924131Z","signature_b64":"UwjirMxVa3IS6nVJ+RmwCeqxV0yKyooYKgFvTJ8cKSEvuc+seTMdaCD7ZJZeJVMUjImTxal0LmNq/7Q3wELLAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14a319a759466f362fed982badd3e48727adcf199d24daf5ec709de4f260363f","last_reissued_at":"2026-07-07T02:18:33.923344Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:18:33.923344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2602.02190","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":"2026-07-07T02:18:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1AEdBCpBKvv0yiFf+eUDUxupP6UbwuyL7d8J6S5K1fb/z4spJ9rOJ39UTcLaOlsABD+U2KsFl8RvqoGHcr2RAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T03:37:44.726264Z"},"content_sha256":"8d3414bb359d55ffdbca98b7827532ab05cff680c9f112fa4ecd201a592fa880","schema_version":"1.0","event_id":"sha256:8d3414bb359d55ffdbca98b7827532ab05cff680c9f112fa4ecd201a592fa880"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:CSRRTJ2ZIZXTML7NTAV23U7EQ4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"PCA of probability measures: Sparse and Dense sampling regimes","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"Elsa Cazelles, Gachon Erell, J\\'er\\'emie Bigot","submitted_at":"2026-02-02T14:56:58Z","abstract_excerpt":"A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-\\alpha}$ for the empirical covariance operator and the PCA excess risk, where $\\alpha>0"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.02190","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/2602.02190/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":"2026-07-07T02:18:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jfqbLjiKS+EeU1IQDNVqGhluPRdKfDvymaPBdK7EmQOOZlJnwRqLITkxlCzS90DNAYGRNpLGgjFuvTRN9sK2BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T03:37:44.727209Z"},"content_sha256":"971724c4184b43216c6c00b9594955c918018ab8c2ec5be935c4e6a9a9b0b506","schema_version":"1.0","event_id":"sha256:971724c4184b43216c6c00b9594955c918018ab8c2ec5be935c4e6a9a9b0b506"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CSRRTJ2ZIZXTML7NTAV23U7EQ4/bundle.json","state_url":"https://pith.science/pith/CSRRTJ2ZIZXTML7NTAV23U7EQ4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CSRRTJ2ZIZXTML7NTAV23U7EQ4/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":"2026-08-08T03:37:44Z","links":{"resolver":"https://pith.science/pith/CSRRTJ2ZIZXTML7NTAV23U7EQ4","bundle":"https://pith.science/pith/CSRRTJ2ZIZXTML7NTAV23U7EQ4/bundle.json","state":"https://pith.science/pith/CSRRTJ2ZIZXTML7NTAV23U7EQ4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CSRRTJ2ZIZXTML7NTAV23U7EQ4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:CSRRTJ2ZIZXTML7NTAV23U7EQ4","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":"f6ddc5ea32cbb5f23eed8812d2983a8c19153bc0964adfce6b1ded0b3bc3e697","cross_cats_sorted":["cs.LG"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"stat.ML","submitted_at":"2026-02-02T14:56:58Z","title_canon_sha256":"63d32ba623176300dfe4ab9ac2e34cac0a2eb2dc86f82f9b62ea95327ac1b1b9"},"schema_version":"1.0","source":{"id":"2602.02190","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.02190","created_at":"2026-07-07T02:18:33Z"},{"alias_kind":"arxiv_version","alias_value":"2602.02190v2","created_at":"2026-07-07T02:18:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.02190","created_at":"2026-07-07T02:18:33Z"},{"alias_kind":"pith_short_12","alias_value":"CSRRTJ2ZIZXT","created_at":"2026-07-07T02:18:33Z"},{"alias_kind":"pith_short_16","alias_value":"CSRRTJ2ZIZXTML7N","created_at":"2026-07-07T02:18:33Z"},{"alias_kind":"pith_short_8","alias_value":"CSRRTJ2Z","created_at":"2026-07-07T02:18:33Z"}],"graph_snapshots":[{"event_id":"sha256:971724c4184b43216c6c00b9594955c918018ab8c2ec5be935c4e6a9a9b0b506","target":"graph","created_at":"2026-07-07T02:18:33Z","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/2602.02190/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-\\alpha}$ for the empirical covariance operator and the PCA excess risk, where $\\alpha>0","authors_text":"Elsa Cazelles, Gachon Erell, J\\'er\\'emie Bigot","cross_cats":["cs.LG"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"stat.ML","submitted_at":"2026-02-02T14:56:58Z","title":"PCA of probability measures: Sparse and Dense sampling regimes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.02190","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":"sha256:8d3414bb359d55ffdbca98b7827532ab05cff680c9f112fa4ecd201a592fa880","target":"record","created_at":"2026-07-07T02:18:33Z","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":"f6ddc5ea32cbb5f23eed8812d2983a8c19153bc0964adfce6b1ded0b3bc3e697","cross_cats_sorted":["cs.LG"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"stat.ML","submitted_at":"2026-02-02T14:56:58Z","title_canon_sha256":"63d32ba623176300dfe4ab9ac2e34cac0a2eb2dc86f82f9b62ea95327ac1b1b9"},"schema_version":"1.0","source":{"id":"2602.02190","kind":"arxiv","version":2}},"canonical_sha256":"14a319a759466f362fed982badd3e48727adcf199d24daf5ec709de4f260363f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14a319a759466f362fed982badd3e48727adcf199d24daf5ec709de4f260363f","first_computed_at":"2026-07-07T02:18:33.923344Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T02:18:33.923344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UwjirMxVa3IS6nVJ+RmwCeqxV0yKyooYKgFvTJ8cKSEvuc+seTMdaCD7ZJZeJVMUjImTxal0LmNq/7Q3wELLAQ==","signature_status":"signed_v1","signed_at":"2026-07-07T02:18:33.924131Z","signed_message":"canonical_sha256_bytes"},"source_id":"2602.02190","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8d3414bb359d55ffdbca98b7827532ab05cff680c9f112fa4ecd201a592fa880","sha256:971724c4184b43216c6c00b9594955c918018ab8c2ec5be935c4e6a9a9b0b506"],"state_sha256":"da054a02b22ddaec4bfe4f53b0775ad414684ba02607c38edafda94a8225301f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VUtI9Tq8UbnorrAWUjHiI+n9jJptxXYjZ6CkiFUBmXE75wCN9T5ddfDH9pOmY+KgGUDnI+J/NxHw8UebatbkCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T03:37:44.732929Z","bundle_sha256":"4aee5154e456222e7a966292c0f57ea46fd4fcb014998e2a41e2bab7a20f0376"}}