{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:HO2RKG43OULBNJDN37C2X6GONG","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":"fa4101861b9b6810f5c69508a8d8182db16675f332a663214f2aeda65cc8fc64","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-03-10T01:56:45Z","title_canon_sha256":"a4adee2a01db35fb81e62a0ef6dce8d3a974ba02f058487aa6d632fde13cd061"},"schema_version":"1.0","source":{"id":"2503.06838","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2503.06838","created_at":"2026-07-05T10:27:43Z"},{"alias_kind":"arxiv_version","alias_value":"2503.06838v1","created_at":"2026-07-05T10:27:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2503.06838","created_at":"2026-07-05T10:27:43Z"},{"alias_kind":"pith_short_12","alias_value":"HO2RKG43OULB","created_at":"2026-07-05T10:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"HO2RKG43OULBNJDN","created_at":"2026-07-05T10:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"HO2RKG43","created_at":"2026-07-05T10:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:60c4cb11ae15eb7a4ebafdac4fca23829007bebf85641da2ff00e2b84b9d894f","target":"graph","created_at":"2026-07-05T10:27:43Z","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/2503.06838/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Suppose we are given two metric spaces and a family of continuous transformations from one to the other. Given a probability distribution on each of these two spaces - namely the source and the target measures - the Wasserstein alignment problem seeks the transformation that minimizes the optimal transport cost between its pushforward of the source distribution and the target distribution, ensuring the closest possible alignment in a probabilistic sense. Examples of interest include two distributions on two Euclidean spaces $\\mathbb{R}^n$ and $\\mathbb{R}^d$, and we want a spatial embedding of ","authors_text":"Bodhisattva Sen, Soumik Pal, Ting-Kam Leonard Wong","cross_cats":["math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-03-10T01:56:45Z","title":"On the Wasserstein alignment problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2503.06838","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:a9b6f10caf07c077ed4cc0781457bae3ac54e79646727689a618c34190ca37de","target":"record","created_at":"2026-07-05T10:27:43Z","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":"fa4101861b9b6810f5c69508a8d8182db16675f332a663214f2aeda65cc8fc64","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-03-10T01:56:45Z","title_canon_sha256":"a4adee2a01db35fb81e62a0ef6dce8d3a974ba02f058487aa6d632fde13cd061"},"schema_version":"1.0","source":{"id":"2503.06838","kind":"arxiv","version":1}},"canonical_sha256":"3bb5151b9b751616a46ddfc5abf8ce699e07463ffe1736e4ca1e1decc5bea067","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3bb5151b9b751616a46ddfc5abf8ce699e07463ffe1736e4ca1e1decc5bea067","first_computed_at":"2026-07-05T10:27:43.946901Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:27:43.946901Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sa2ZTeXjMU9ngCwKMTznLCYYTVYheTK/dklMfB6pq1YWQX5ePR09VJHM5z+GsJGP5U3uZlLekHMIpnZccokPDw==","signature_status":"signed_v1","signed_at":"2026-07-05T10:27:43.947523Z","signed_message":"canonical_sha256_bytes"},"source_id":"2503.06838","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a9b6f10caf07c077ed4cc0781457bae3ac54e79646727689a618c34190ca37de","sha256:60c4cb11ae15eb7a4ebafdac4fca23829007bebf85641da2ff00e2b84b9d894f"],"state_sha256":"4d0b0360b7055e06ab0438d8e6ee440fa22e64d9a7782f72ebf8de5d2d8c5834"}