{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:OS73PTWPJAMIEVGTNRX4AMGQBN","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":"b5be0c52d7a11c427f168f8bafe52487bb68ae1abc72e01ea16c54501375f460","cross_cats_sorted":["cs.ET"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-05-30T07:01:14Z","title_canon_sha256":"56c46ca4b673f88fbf0d25ec50ae809b2b00966d3dd5f7049163497ef9cd2f55"},"schema_version":"1.0","source":{"id":"1405.7786","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.7786","created_at":"2026-05-18T01:20:02Z"},{"alias_kind":"arxiv_version","alias_value":"1405.7786v2","created_at":"2026-05-18T01:20:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.7786","created_at":"2026-05-18T01:20:02Z"},{"alias_kind":"pith_short_12","alias_value":"OS73PTWPJAMI","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_16","alias_value":"OS73PTWPJAMIEVGT","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_8","alias_value":"OS73PTWP","created_at":"2026-05-18T12:28:43Z"}],"graph_snapshots":[{"event_id":"sha256:571ad237dd0193e8fdb25b428e1643f5460a5cd774c44a10873ec19857efd617","target":"graph","created_at":"2026-05-18T01:20:02Z","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 discuss extended definitions of linear and multilinear operations such as Kronecker, Hadamard, and contracted products, and establish links between them for tensor calculus. Then we introduce effective low-rank tensor approximation techniques including Candecomp/Parafac (CP), Tucker, and tensor train (TT) decompositions with a number of mathematical and graphical representations. We also provide a brief review of mathematical properties of the TT decomposition as a low-rank approximation technique. With the aim of breaking the curse-of-dimensionality in large-scale numerical analysis, we de","authors_text":"Andrzej Cichocki, Namgil Lee","cross_cats":["cs.ET"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-05-30T07:01:14Z","title":"Fundamental Tensor Operations for Large-Scale Data Analysis in Tensor Train Formats"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.7786","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":"sha256:3cb63fa9010bf070a4cd74c7ad18de15afd12a4a12157c6e669a136dcc151216","target":"record","created_at":"2026-05-18T01:20:02Z","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":"b5be0c52d7a11c427f168f8bafe52487bb68ae1abc72e01ea16c54501375f460","cross_cats_sorted":["cs.ET"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2014-05-30T07:01:14Z","title_canon_sha256":"56c46ca4b673f88fbf0d25ec50ae809b2b00966d3dd5f7049163497ef9cd2f55"},"schema_version":"1.0","source":{"id":"1405.7786","kind":"arxiv","version":2}},"canonical_sha256":"74bfb7cecf48188254d36c6fc030d00b6b1aa1d80e2f4d90b780159755b8cb59","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"74bfb7cecf48188254d36c6fc030d00b6b1aa1d80e2f4d90b780159755b8cb59","first_computed_at":"2026-05-18T01:20:02.046160Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:02.046160Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YLJIqouSe0NM0VbHAW74CTXJz2UlWQtM3fpNh2A0ka/GfKC2F1CwvY3cI+dQusvjRUmxafEKZb2M8kUGTJ5HCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:02.046933Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.7786","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3cb63fa9010bf070a4cd74c7ad18de15afd12a4a12157c6e669a136dcc151216","sha256:571ad237dd0193e8fdb25b428e1643f5460a5cd774c44a10873ec19857efd617"],"state_sha256":"80b4e57fd37a30e89871e9d42561838513b67ad81336611fbaf8726acaf0146f"}