{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:NDOOWDKW5IY2P5VVHRF7PSRXHE","short_pith_number":"pith:NDOOWDKW","canonical_record":{"source":{"id":"1809.04249","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-12T04:13:48Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"a8a5ce2d7dec335395043cebc8047b5c396b819d0c8cc3d1db9d5d0250ebac9e","abstract_canon_sha256":"9ab3817c62c0e1a0bacd06b6dc71c584315b442ae4d82b27eb35ffa666777c1a"},"schema_version":"1.0"},"canonical_sha256":"68dceb0d56ea31a7f6b53c4bf7ca3739082c0711f9c5fca4bbc4b664c9a32dfc","source":{"kind":"arxiv","id":"1809.04249","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.04249","created_at":"2026-07-05T02:01:59Z"},{"alias_kind":"arxiv_version","alias_value":"1809.04249v5","created_at":"2026-07-05T02:01:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.04249","created_at":"2026-07-05T02:01:59Z"},{"alias_kind":"pith_short_12","alias_value":"NDOOWDKW5IY2","created_at":"2026-07-05T02:01:59Z"},{"alias_kind":"pith_short_16","alias_value":"NDOOWDKW5IY2P5VV","created_at":"2026-07-05T02:01:59Z"},{"alias_kind":"pith_short_8","alias_value":"NDOOWDKW","created_at":"2026-07-05T02:01:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:NDOOWDKW5IY2P5VVHRF7PSRXHE","target":"record","payload":{"canonical_record":{"source":{"id":"1809.04249","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-12T04:13:48Z","cross_cats_sorted":["stat.ML"],"title_canon_sha256":"a8a5ce2d7dec335395043cebc8047b5c396b819d0c8cc3d1db9d5d0250ebac9e","abstract_canon_sha256":"9ab3817c62c0e1a0bacd06b6dc71c584315b442ae4d82b27eb35ffa666777c1a"},"schema_version":"1.0"},"canonical_sha256":"68dceb0d56ea31a7f6b53c4bf7ca3739082c0711f9c5fca4bbc4b664c9a32dfc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:01:59.645366Z","signature_b64":"BHxfjNcxdsnZwp/+VKRxM6LcxlhYQNaLM9JJ8tZV50YxuatyMmG5X1Z1wsAWJKUltVWhTjvuwv2CNSIOmNC/AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"68dceb0d56ea31a7f6b53c4bf7ca3739082c0711f9c5fca4bbc4b664c9a32dfc","last_reissued_at":"2026-07-05T02:01:59.644972Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:01:59.644972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.04249","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:01:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6bSrkz+J2q7ZeCxOFUzPdFlfKTBy8hKpmPunklhUguY+w0ssUbpTaU6uLrp718Tw7UZH22hC2v86gYLsirIrBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T02:26:10.894084Z"},"content_sha256":"0b6d5617aab1d6e2de60a1ed880187ffcc89f1fff21982b62411e24113d79b0e","schema_version":"1.0","event_id":"sha256:0b6d5617aab1d6e2de60a1ed880187ffcc89f1fff21982b62411e24113d79b0e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:NDOOWDKW5IY2P5VVHRF7PSRXHE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Fast Globally Linearly Convergent Algorithm for the Computation of Wasserstein Barycenters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML"],"primary_cat":"math.OC","authors_text":"Defeng Sun, Jia Li, Kim-Chuan Toh, Lei Yang","submitted_at":"2018-09-12T04:13:48Z","abstract_excerpt":"We consider the problem of computing a Wasserstein barycenter for a set of discrete probability distributions with finite supports, which finds many applications in areas such as statistics, machine learning and image processing. When the support points of the barycenter are pre-specified, this problem can be modeled as a linear programming (LP) problem whose size can be extremely large. To handle this large-scale LP, we analyse the structure of its dual problem, which is conceivably more tractable and can be reformulated as a well-structured convex problem with 3 kinds of block variables and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.04249","kind":"arxiv","version":5},"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/1809.04249/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":"2026-07-05T02:01:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c2u6hraufdDaEnVowF2HaQdiDEbtKzOKcyQgFM0lNr37fuGBULXunoysmdaaCwsRfn/sWoG5P9P6AXOm0HEPDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T02:26:10.894478Z"},"content_sha256":"0dc5de79cc59dcd1e6506f360bcecc4f5f1d2d0ff67784557a5ce7e29eb3cae9","schema_version":"1.0","event_id":"sha256:0dc5de79cc59dcd1e6506f360bcecc4f5f1d2d0ff67784557a5ce7e29eb3cae9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NDOOWDKW5IY2P5VVHRF7PSRXHE/bundle.json","state_url":"https://pith.science/pith/NDOOWDKW5IY2P5VVHRF7PSRXHE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NDOOWDKW5IY2P5VVHRF7PSRXHE/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":"2026-08-14T02:26:10Z","links":{"resolver":"https://pith.science/pith/NDOOWDKW5IY2P5VVHRF7PSRXHE","bundle":"https://pith.science/pith/NDOOWDKW5IY2P5VVHRF7PSRXHE/bundle.json","state":"https://pith.science/pith/NDOOWDKW5IY2P5VVHRF7PSRXHE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NDOOWDKW5IY2P5VVHRF7PSRXHE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:NDOOWDKW5IY2P5VVHRF7PSRXHE","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":"9ab3817c62c0e1a0bacd06b6dc71c584315b442ae4d82b27eb35ffa666777c1a","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-12T04:13:48Z","title_canon_sha256":"a8a5ce2d7dec335395043cebc8047b5c396b819d0c8cc3d1db9d5d0250ebac9e"},"schema_version":"1.0","source":{"id":"1809.04249","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.04249","created_at":"2026-07-05T02:01:59Z"},{"alias_kind":"arxiv_version","alias_value":"1809.04249v5","created_at":"2026-07-05T02:01:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.04249","created_at":"2026-07-05T02:01:59Z"},{"alias_kind":"pith_short_12","alias_value":"NDOOWDKW5IY2","created_at":"2026-07-05T02:01:59Z"},{"alias_kind":"pith_short_16","alias_value":"NDOOWDKW5IY2P5VV","created_at":"2026-07-05T02:01:59Z"},{"alias_kind":"pith_short_8","alias_value":"NDOOWDKW","created_at":"2026-07-05T02:01:59Z"}],"graph_snapshots":[{"event_id":"sha256:0dc5de79cc59dcd1e6506f360bcecc4f5f1d2d0ff67784557a5ce7e29eb3cae9","target":"graph","created_at":"2026-07-05T02:01:59Z","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/1809.04249/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider the problem of computing a Wasserstein barycenter for a set of discrete probability distributions with finite supports, which finds many applications in areas such as statistics, machine learning and image processing. When the support points of the barycenter are pre-specified, this problem can be modeled as a linear programming (LP) problem whose size can be extremely large. To handle this large-scale LP, we analyse the structure of its dual problem, which is conceivably more tractable and can be reformulated as a well-structured convex problem with 3 kinds of block variables and ","authors_text":"Defeng Sun, Jia Li, Kim-Chuan Toh, Lei Yang","cross_cats":["stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-12T04:13:48Z","title":"A Fast Globally Linearly Convergent Algorithm for the Computation of Wasserstein Barycenters"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.04249","kind":"arxiv","version":5},"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":"sha256:0b6d5617aab1d6e2de60a1ed880187ffcc89f1fff21982b62411e24113d79b0e","target":"record","created_at":"2026-07-05T02:01:59Z","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":"9ab3817c62c0e1a0bacd06b6dc71c584315b442ae4d82b27eb35ffa666777c1a","cross_cats_sorted":["stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-12T04:13:48Z","title_canon_sha256":"a8a5ce2d7dec335395043cebc8047b5c396b819d0c8cc3d1db9d5d0250ebac9e"},"schema_version":"1.0","source":{"id":"1809.04249","kind":"arxiv","version":5}},"canonical_sha256":"68dceb0d56ea31a7f6b53c4bf7ca3739082c0711f9c5fca4bbc4b664c9a32dfc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68dceb0d56ea31a7f6b53c4bf7ca3739082c0711f9c5fca4bbc4b664c9a32dfc","first_computed_at":"2026-07-05T02:01:59.644972Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:01:59.644972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BHxfjNcxdsnZwp/+VKRxM6LcxlhYQNaLM9JJ8tZV50YxuatyMmG5X1Z1wsAWJKUltVWhTjvuwv2CNSIOmNC/AQ==","signature_status":"signed_v1","signed_at":"2026-07-05T02:01:59.645366Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.04249","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0b6d5617aab1d6e2de60a1ed880187ffcc89f1fff21982b62411e24113d79b0e","sha256:0dc5de79cc59dcd1e6506f360bcecc4f5f1d2d0ff67784557a5ce7e29eb3cae9"],"state_sha256":"6abd14b597d3b19d6d359553a25d86fd3c952bc260673bfd5054fdfd94168748"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yrmQkHslFSewoKTaqfX3lWxSNdaGLmrc67UmcjL1uh3LwhoiwxmwleEJbmK+sZ3arPllT3wiZdtDN1uXmDppBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T02:26:10.897475Z","bundle_sha256":"353b43d940e33a0fbbcc7ec0baae4613c98734739302f521dc53b7e759cb7ac1"}}