{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:EOJNVK5KRVMQ6PL6F4UBFL5T4Q","short_pith_number":"pith:EOJNVK5K","schema_version":"1.0","canonical_sha256":"2392daabaa8d590f3d7e2f2812afb3e41642ae0c075e9ce997264412650109a5","source":{"kind":"arxiv","id":"1505.03027","version":2},"attestation_state":"computed","paper":{"title":"Geometric Structures in Tensor Representations (Final Release)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.FA"],"primary_cat":"math.NA","authors_text":"Anthony Nouy, Antonio Falco, Wolfgang Hackbusch","submitted_at":"2015-05-12T14:38:36Z","abstract_excerpt":"The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the dimensions of the associated minimal subspaces, we introduce, in the underlying algebraic tensor space, the set of tensors in a tree-based format with either bounded or fixed tree-based rank. This class contains the Tucker format and the Hierarchical Tucker format (including the T"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1505.03027","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-05-12T14:38:36Z","cross_cats_sorted":["math.DG","math.FA"],"title_canon_sha256":"ba5bde0339d5d3e4049122d1b67c8520e41a0c1ac9b4a2747545213d008549d5","abstract_canon_sha256":"0f3f4fe23a6491a61bcee9ef19ca77a18da135e29b4ebaefb5ce8010ef238034"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:41:45.961337Z","signature_b64":"OlQhjkVVe7yFznWQEOhjFsyO0yqulGDizXnXUaosP66yyILVSWxlJuAYGb0QI2y474Tk17TrvNwMxEmurmGtDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2392daabaa8d590f3d7e2f2812afb3e41642ae0c075e9ce997264412650109a5","last_reissued_at":"2026-05-18T01:41:45.960876Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:41:45.960876Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometric Structures in Tensor Representations (Final Release)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.FA"],"primary_cat":"math.NA","authors_text":"Anthony Nouy, Antonio Falco, Wolfgang Hackbusch","submitted_at":"2015-05-12T14:38:36Z","abstract_excerpt":"The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the dimensions of the associated minimal subspaces, we introduce, in the underlying algebraic tensor space, the set of tensors in a tree-based format with either bounded or fixed tree-based rank. This class contains the Tucker format and the Hierarchical Tucker format (including the T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03027","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1505.03027","created_at":"2026-05-18T01:41:45.960944+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.03027v2","created_at":"2026-05-18T01:41:45.960944+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.03027","created_at":"2026-05-18T01:41:45.960944+00:00"},{"alias_kind":"pith_short_12","alias_value":"EOJNVK5KRVMQ","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_16","alias_value":"EOJNVK5KRVMQ6PL6","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_8","alias_value":"EOJNVK5K","created_at":"2026-05-18T12:29:19.899920+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2504.15516","citing_title":"Derivation of Runge--Kutta Order Conditions via Functional Tree Tensor Networks","ref_index":23,"is_internal_anchor":true},{"citing_arxiv_id":"2604.05890","citing_title":"A Tensor-Train Framework for Bayesian Inference in High-Dimensional Systems: Applications to MIMO Detection and Channel Decoding","ref_index":38,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EOJNVK5KRVMQ6PL6F4UBFL5T4Q","json":"https://pith.science/pith/EOJNVK5KRVMQ6PL6F4UBFL5T4Q.json","graph_json":"https://pith.science/api/pith-number/EOJNVK5KRVMQ6PL6F4UBFL5T4Q/graph.json","events_json":"https://pith.science/api/pith-number/EOJNVK5KRVMQ6PL6F4UBFL5T4Q/events.json","paper":"https://pith.science/paper/EOJNVK5K"},"agent_actions":{"view_html":"https://pith.science/pith/EOJNVK5KRVMQ6PL6F4UBFL5T4Q","download_json":"https://pith.science/pith/EOJNVK5KRVMQ6PL6F4UBFL5T4Q.json","view_paper":"https://pith.science/paper/EOJNVK5K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.03027&json=true","fetch_graph":"https://pith.science/api/pith-number/EOJNVK5KRVMQ6PL6F4UBFL5T4Q/graph.json","fetch_events":"https://pith.science/api/pith-number/EOJNVK5KRVMQ6PL6F4UBFL5T4Q/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EOJNVK5KRVMQ6PL6F4UBFL5T4Q/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EOJNVK5KRVMQ6PL6F4UBFL5T4Q/action/storage_attestation","attest_author":"https://pith.science/pith/EOJNVK5KRVMQ6PL6F4UBFL5T4Q/action/author_attestation","sign_citation":"https://pith.science/pith/EOJNVK5KRVMQ6PL6F4UBFL5T4Q/action/citation_signature","submit_replication":"https://pith.science/pith/EOJNVK5KRVMQ6PL6F4UBFL5T4Q/action/replication_record"}},"created_at":"2026-05-18T01:41:45.960944+00:00","updated_at":"2026-05-18T01:41:45.960944+00:00"}