{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:WCVWYOK624AYQLCPAJKGAJMQDE","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":"2a0d8326f1168f2be97066d66df6735eb7412965bfa0725f345aa50025a779bf","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-07-03T12:57:57Z","title_canon_sha256":"faeaaddafb0dc42225f7e713ecad7bc908601ebed943c4aeff7211f339e712fb"},"schema_version":"1.0","source":{"id":"1407.0900","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1407.0900","created_at":"2026-05-18T01:12:22Z"},{"alias_kind":"arxiv_version","alias_value":"1407.0900v3","created_at":"2026-05-18T01:12:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.0900","created_at":"2026-05-18T01:12:22Z"},{"alias_kind":"pith_short_12","alias_value":"WCVWYOK624AY","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_16","alias_value":"WCVWYOK624AYQLCP","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_8","alias_value":"WCVWYOK6","created_at":"2026-05-18T12:28:54Z"}],"graph_snapshots":[{"event_id":"sha256:2f998d8f7a32d4a12d3f2c95540cf973e908aa2d3011fc434fe620c193fcb33f","target":"graph","created_at":"2026-05-18T01:12:22Z","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"},"paper":{"abstract_excerpt":"We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is given by the distance of a point to a Schubert variety within the Grassmannian. This distance reduces to the Grassmann distance when the subspaces are equidimensional and does not depend on any embedding into a larger ambient space. Furthermore, it has a concrete expression involving principal angles, and is efficiently computable in numerically stable ways.","authors_text":"Ke Ye, Lek-Heng Lim","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-07-03T12:57:57Z","title":"Schubert varieties and distances between subspaces of different dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0900","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:b57a6f466c758b8890c5b9a6a9de7918f05e4bfd97c14e45d9f3d83482ae528f","target":"record","created_at":"2026-05-18T01:12:22Z","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":"2a0d8326f1168f2be97066d66df6735eb7412965bfa0725f345aa50025a779bf","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-07-03T12:57:57Z","title_canon_sha256":"faeaaddafb0dc42225f7e713ecad7bc908601ebed943c4aeff7211f339e712fb"},"schema_version":"1.0","source":{"id":"1407.0900","kind":"arxiv","version":3}},"canonical_sha256":"b0ab6c395ed701882c4f0254602590191558642a2cb03f293642f50d95b9c9e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b0ab6c395ed701882c4f0254602590191558642a2cb03f293642f50d95b9c9e0","first_computed_at":"2026-05-18T01:12:22.984503Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:22.984503Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"19PkVqwA4aTgCdeKlvtg2w175c9L2wePa/HfDIpyZwgG8ULpHj/bU+HcZEeI1kFlNaJeqI6yckBwAyystuQ4CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:22.984923Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.0900","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b57a6f466c758b8890c5b9a6a9de7918f05e4bfd97c14e45d9f3d83482ae528f","sha256:2f998d8f7a32d4a12d3f2c95540cf973e908aa2d3011fc434fe620c193fcb33f"],"state_sha256":"e6ac015b2e03cc53db54cf0170b730a289c816effaad279da3386dfc3a67274a"}