{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:XPDNIMO2H3KIXCFNXLU47DPOFB","short_pith_number":"pith:XPDNIMO2","canonical_record":{"source":{"id":"2106.02538","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2021-06-04T15:07:39Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"47b6735a2ac3f99f83d2a38c79c1ccb1148474eb813ff0b55e857f7fae61c399","abstract_canon_sha256":"ed7720cd0cf24bbf46600c7e889de77fba2d8115c961b46c5b57c775833e2527"},"schema_version":"1.0"},"canonical_sha256":"bbc6d431da3ed48b88adbae9cf8dee28452062243fd24d70d79fc641f524a4a3","source":{"kind":"arxiv","id":"2106.02538","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.02538","created_at":"2026-07-05T09:11:58Z"},{"alias_kind":"arxiv_version","alias_value":"2106.02538v3","created_at":"2026-07-05T09:11:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.02538","created_at":"2026-07-05T09:11:58Z"},{"alias_kind":"pith_short_12","alias_value":"XPDNIMO2H3KI","created_at":"2026-07-05T09:11:58Z"},{"alias_kind":"pith_short_16","alias_value":"XPDNIMO2H3KIXCFN","created_at":"2026-07-05T09:11:58Z"},{"alias_kind":"pith_short_8","alias_value":"XPDNIMO2","created_at":"2026-07-05T09:11:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:XPDNIMO2H3KIXCFNXLU47DPOFB","target":"record","payload":{"canonical_record":{"source":{"id":"2106.02538","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2021-06-04T15:07:39Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"47b6735a2ac3f99f83d2a38c79c1ccb1148474eb813ff0b55e857f7fae61c399","abstract_canon_sha256":"ed7720cd0cf24bbf46600c7e889de77fba2d8115c961b46c5b57c775833e2527"},"schema_version":"1.0"},"canonical_sha256":"bbc6d431da3ed48b88adbae9cf8dee28452062243fd24d70d79fc641f524a4a3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:11:58.763789Z","signature_b64":"ZNK5f0hZjeNCcst6d10iVIUe6tv1WEOCG/DyWDWPXB6mVpdfw4UkCLTmO7PdqR1+gMYF6QIUjBfmQCRu6Xh+Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bbc6d431da3ed48b88adbae9cf8dee28452062243fd24d70d79fc641f524a4a3","last_reissued_at":"2026-07-05T09:11:58.763343Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:11:58.763343Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2106.02538","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T09:11:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JLaxDClDNd6tT7C9FmaBfmJi+cY2JdmChrW7wBs9Pwd2uUpAaO8fWuDlKGaLkvc3fLUwAY09UPJnG+JGrsXYDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T04:37:21.380997Z"},"content_sha256":"698085d24d34a4d7b9c01662956bfda2fe8fe0e04b1815242b0d2285a1682caf","schema_version":"1.0","event_id":"sha256:698085d24d34a4d7b9c01662956bfda2fe8fe0e04b1815242b0d2285a1682caf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:XPDNIMO2H3KIXCFNXLU47DPOFB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bottleneck Profiles and Discrete Prokhorov Metrics for Persistence Diagrams","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"cs.CG","authors_text":"Niklas Hellmer, Pawe{\\l} D{\\l}otko","submitted_at":"2021-06-04T15:07:39Z","abstract_excerpt":"In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in a systematic way by introducing bottleneck profiles. This leads to a notion of discrete Prokhorov metrics for persistence diagrams as a generalization of the Bottleneck distance. They satisfy a stability result and bounds with respect to Wasserstein metrics. We provide algorithms to compute the newly introduced quantities and end with an discussion about ex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.02538","kind":"arxiv","version":3},"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/2106.02538/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":"2026-07-05T09:11:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xQaga4eCzT1azfVJ8PkVpgsJkFdLY50dcZFXa56StxeZfk7o69IpdTJz1/6K4vQktwvZSK/9LhcG3xVrwWisDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T04:37:21.381382Z"},"content_sha256":"7ffa252ea49456ef92ce3e1a0b88564b23b7accec161af8cca8697254f3b993b","schema_version":"1.0","event_id":"sha256:7ffa252ea49456ef92ce3e1a0b88564b23b7accec161af8cca8697254f3b993b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XPDNIMO2H3KIXCFNXLU47DPOFB/bundle.json","state_url":"https://pith.science/pith/XPDNIMO2H3KIXCFNXLU47DPOFB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XPDNIMO2H3KIXCFNXLU47DPOFB/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":"2026-08-20T04:37:21Z","links":{"resolver":"https://pith.science/pith/XPDNIMO2H3KIXCFNXLU47DPOFB","bundle":"https://pith.science/pith/XPDNIMO2H3KIXCFNXLU47DPOFB/bundle.json","state":"https://pith.science/pith/XPDNIMO2H3KIXCFNXLU47DPOFB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XPDNIMO2H3KIXCFNXLU47DPOFB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:XPDNIMO2H3KIXCFNXLU47DPOFB","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":"ed7720cd0cf24bbf46600c7e889de77fba2d8115c961b46c5b57c775833e2527","cross_cats_sorted":["math.AT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2021-06-04T15:07:39Z","title_canon_sha256":"47b6735a2ac3f99f83d2a38c79c1ccb1148474eb813ff0b55e857f7fae61c399"},"schema_version":"1.0","source":{"id":"2106.02538","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.02538","created_at":"2026-07-05T09:11:58Z"},{"alias_kind":"arxiv_version","alias_value":"2106.02538v3","created_at":"2026-07-05T09:11:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.02538","created_at":"2026-07-05T09:11:58Z"},{"alias_kind":"pith_short_12","alias_value":"XPDNIMO2H3KI","created_at":"2026-07-05T09:11:58Z"},{"alias_kind":"pith_short_16","alias_value":"XPDNIMO2H3KIXCFN","created_at":"2026-07-05T09:11:58Z"},{"alias_kind":"pith_short_8","alias_value":"XPDNIMO2","created_at":"2026-07-05T09:11:58Z"}],"graph_snapshots":[{"event_id":"sha256:7ffa252ea49456ef92ce3e1a0b88564b23b7accec161af8cca8697254f3b993b","target":"graph","created_at":"2026-07-05T09:11:58Z","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/2106.02538/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in a systematic way by introducing bottleneck profiles. This leads to a notion of discrete Prokhorov metrics for persistence diagrams as a generalization of the Bottleneck distance. They satisfy a stability result and bounds with respect to Wasserstein metrics. We provide algorithms to compute the newly introduced quantities and end with an discussion about ex","authors_text":"Niklas Hellmer, Pawe{\\l} D{\\l}otko","cross_cats":["math.AT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2021-06-04T15:07:39Z","title":"Bottleneck Profiles and Discrete Prokhorov Metrics for Persistence Diagrams"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.02538","kind":"arxiv","version":3},"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":"sha256:698085d24d34a4d7b9c01662956bfda2fe8fe0e04b1815242b0d2285a1682caf","target":"record","created_at":"2026-07-05T09:11:58Z","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":"ed7720cd0cf24bbf46600c7e889de77fba2d8115c961b46c5b57c775833e2527","cross_cats_sorted":["math.AT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CG","submitted_at":"2021-06-04T15:07:39Z","title_canon_sha256":"47b6735a2ac3f99f83d2a38c79c1ccb1148474eb813ff0b55e857f7fae61c399"},"schema_version":"1.0","source":{"id":"2106.02538","kind":"arxiv","version":3}},"canonical_sha256":"bbc6d431da3ed48b88adbae9cf8dee28452062243fd24d70d79fc641f524a4a3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bbc6d431da3ed48b88adbae9cf8dee28452062243fd24d70d79fc641f524a4a3","first_computed_at":"2026-07-05T09:11:58.763343Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:11:58.763343Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZNK5f0hZjeNCcst6d10iVIUe6tv1WEOCG/DyWDWPXB6mVpdfw4UkCLTmO7PdqR1+gMYF6QIUjBfmQCRu6Xh+Ag==","signature_status":"signed_v1","signed_at":"2026-07-05T09:11:58.763789Z","signed_message":"canonical_sha256_bytes"},"source_id":"2106.02538","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:698085d24d34a4d7b9c01662956bfda2fe8fe0e04b1815242b0d2285a1682caf","sha256:7ffa252ea49456ef92ce3e1a0b88564b23b7accec161af8cca8697254f3b993b"],"state_sha256":"28ac8782688b1108ba7a348e527f32676eaf4095cc85a01a0b6d875d51bb4d14"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sWydpbO/jdDDCYGdpX80hGPt0aEg8+nSK3sCJ9D4yafzJpoEi/D3QCOX6KqAyyyB8flZloEYBqsb5eiJ6NEuCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T04:37:21.383701Z","bundle_sha256":"4daae976d3904fba5d2413ca7bc0e22b754a1355464c312cfde1c86506819f6b"}}