{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:F7QNJPFUUIKKKWMWAIBQUWILNU","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":"fa3d2ac7973490b59d70b75e540183ab0e8b80d476e249966eb2ed91b5ce1f27","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-09-29T07:50:25Z","title_canon_sha256":"21503b89fb522c2134e27af2ae63d58b15b53ebe162f9d016b017510e62eb342"},"schema_version":"1.0","source":{"id":"1810.01262","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.01262","created_at":"2026-06-04T20:13:24Z"},{"alias_kind":"arxiv_version","alias_value":"1810.01262v2","created_at":"2026-06-04T20:13:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.01262","created_at":"2026-06-04T20:13:24Z"},{"alias_kind":"pith_short_12","alias_value":"F7QNJPFUUIKK","created_at":"2026-06-04T20:13:24Z"},{"alias_kind":"pith_short_16","alias_value":"F7QNJPFUUIKKKWMW","created_at":"2026-06-04T20:13:24Z"},{"alias_kind":"pith_short_8","alias_value":"F7QNJPFU","created_at":"2026-06-04T20:13:24Z"}],"graph_snapshots":[{"event_id":"sha256:92f0a6cf5b5bf44abbe7b43efc305d9d849ed5bdc5447d35bdc8b3758b7f9184","target":"graph","created_at":"2026-06-04T20:13:24Z","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/1810.01262/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The main goal of this paper is to study the topological properties of tensors in tree-based Tucker format. These formats include the Tucker format and the Hierarchical Tucker format. A property of the so-called minimal subspaces is used for obtaining a representation of tensors with either bounded or fixed tree-based rank in the underlying algebraic tensor space. We provide a new characterisation of minimal subspaces which extends the existing characterisations. We also introduce a definition of topological tensor spaces in tree-based format, with the introduction of a norm at each vertex of t","authors_text":"Anthony Nouy, Antonio Falco, Wolfgang Hackbusch","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-09-29T07:50:25Z","title":"Tree-based tensor formats"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.01262","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:a99170a2f729a3a0a42ee8a2144372d119cffd5a9a9b9b44ee33fa79b9e9e46f","target":"record","created_at":"2026-06-04T20:13:24Z","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":"fa3d2ac7973490b59d70b75e540183ab0e8b80d476e249966eb2ed91b5ce1f27","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2018-09-29T07:50:25Z","title_canon_sha256":"21503b89fb522c2134e27af2ae63d58b15b53ebe162f9d016b017510e62eb342"},"schema_version":"1.0","source":{"id":"1810.01262","kind":"arxiv","version":2}},"canonical_sha256":"2fe0d4bcb4a214a5599602030a590b6d04d89906a2660b0992f7355013a28d9a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2fe0d4bcb4a214a5599602030a590b6d04d89906a2660b0992f7355013a28d9a","first_computed_at":"2026-06-04T20:13:24.697814Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-04T20:13:24.697814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"b4PG/R/lO4JU4KWP2LjkeljNLLuXtmlm5QFT/l7gK/34Ntzldueix5yTeHxPcg1GZKMx48n5sEeJy8eP+HbtDA==","signature_status":"signed_v1","signed_at":"2026-06-04T20:13:24.698371Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.01262","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a99170a2f729a3a0a42ee8a2144372d119cffd5a9a9b9b44ee33fa79b9e9e46f","sha256:92f0a6cf5b5bf44abbe7b43efc305d9d849ed5bdc5447d35bdc8b3758b7f9184"],"state_sha256":"def30dae2a482fe6ee36ea90390b52f6371b279227ecffb639618c68241f00d4"}