{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:LOIPSQYMT6WERIIEAGOGBHU745","short_pith_number":"pith:LOIPSQYM","schema_version":"1.0","canonical_sha256":"5b90f9430c9fac48a104019c609e9fe74fde11604a225c4270e3fe47c7eae137","source":{"kind":"arxiv","id":"2403.00643","version":2},"attestation_state":"computed","paper":{"title":"Undercomplete Decomposition of Symmetric Tensors in Linear Time, and Smoothed Analysis of the Condition Number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.NA","math.NA"],"primary_cat":"cs.DS","authors_text":"Pascal Koiran, Subhayan Saha","submitted_at":"2024-03-01T16:22:39Z","abstract_excerpt":"We study symmetric tensor decompositions, i.e., decompositions of the form $T = \\sum_{i=1}^r u_i^{\\otimes 3}$ where $T$ is a symmetric tensor of order 3 and $u_i \\in \\mathbb{C}^n$.In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from $u_i$. In this paper we assume that the $u_i$ are linearly independent.This implies $r \\leq n$,that is, the decomposition of T is undercomplete.\n  We give a randomized algorithm for the following problem in the exact arithmetic model of computation: Let $T$ be an order-3 symmetric tensor that has an undercompl"},"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":"2403.00643","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2024-03-01T16:22:39Z","cross_cats_sorted":["cs.CC","cs.NA","math.NA"],"title_canon_sha256":"733feaf812e7d8d054659f1d6e2801e62c6cc125b52ade649deebf6642d7b735","abstract_canon_sha256":"1de79f518ee993c203203cdf9179db9f7891f28538721e7133ae98804a3fc00d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:28:14.758026Z","signature_b64":"BXQB0aWrlpyAQk5HuySlq14TQXYvfdWdnx21ggsITV+nsjSTJ9OMZA4EFrb9os74H9Tad9K1LRWmitQ4MqELCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b90f9430c9fac48a104019c609e9fe74fde11604a225c4270e3fe47c7eae137","last_reissued_at":"2026-07-05T10:28:14.757421Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:28:14.757421Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Undercomplete Decomposition of Symmetric Tensors in Linear Time, and Smoothed Analysis of the Condition Number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.NA","math.NA"],"primary_cat":"cs.DS","authors_text":"Pascal Koiran, Subhayan Saha","submitted_at":"2024-03-01T16:22:39Z","abstract_excerpt":"We study symmetric tensor decompositions, i.e., decompositions of the form $T = \\sum_{i=1}^r u_i^{\\otimes 3}$ where $T$ is a symmetric tensor of order 3 and $u_i \\in \\mathbb{C}^n$.In order to obtain efficient decomposition algorithms, it is necessary to require additional properties from $u_i$. In this paper we assume that the $u_i$ are linearly independent.This implies $r \\leq n$,that is, the decomposition of T is undercomplete.\n  We give a randomized algorithm for the following problem in the exact arithmetic model of computation: Let $T$ be an order-3 symmetric tensor that has an undercompl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.00643","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2403.00643/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2403.00643","created_at":"2026-07-05T10:28:14.757490+00:00"},{"alias_kind":"arxiv_version","alias_value":"2403.00643v2","created_at":"2026-07-05T10:28:14.757490+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.00643","created_at":"2026-07-05T10:28:14.757490+00:00"},{"alias_kind":"pith_short_12","alias_value":"LOIPSQYMT6WE","created_at":"2026-07-05T10:28:14.757490+00:00"},{"alias_kind":"pith_short_16","alias_value":"LOIPSQYMT6WERIIE","created_at":"2026-07-05T10:28:14.757490+00:00"},{"alias_kind":"pith_short_8","alias_value":"LOIPSQYM","created_at":"2026-07-05T10:28:14.757490+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2505.12713","citing_title":"Identifiability of Nonnegative Tucker Decompositions -- Part I: Theory","ref_index":31,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LOIPSQYMT6WERIIEAGOGBHU745","json":"https://pith.science/pith/LOIPSQYMT6WERIIEAGOGBHU745.json","graph_json":"https://pith.science/api/pith-number/LOIPSQYMT6WERIIEAGOGBHU745/graph.json","events_json":"https://pith.science/api/pith-number/LOIPSQYMT6WERIIEAGOGBHU745/events.json","paper":"https://pith.science/paper/LOIPSQYM"},"agent_actions":{"view_html":"https://pith.science/pith/LOIPSQYMT6WERIIEAGOGBHU745","download_json":"https://pith.science/pith/LOIPSQYMT6WERIIEAGOGBHU745.json","view_paper":"https://pith.science/paper/LOIPSQYM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2403.00643&json=true","fetch_graph":"https://pith.science/api/pith-number/LOIPSQYMT6WERIIEAGOGBHU745/graph.json","fetch_events":"https://pith.science/api/pith-number/LOIPSQYMT6WERIIEAGOGBHU745/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LOIPSQYMT6WERIIEAGOGBHU745/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LOIPSQYMT6WERIIEAGOGBHU745/action/storage_attestation","attest_author":"https://pith.science/pith/LOIPSQYMT6WERIIEAGOGBHU745/action/author_attestation","sign_citation":"https://pith.science/pith/LOIPSQYMT6WERIIEAGOGBHU745/action/citation_signature","submit_replication":"https://pith.science/pith/LOIPSQYMT6WERIIEAGOGBHU745/action/replication_record"}},"created_at":"2026-07-05T10:28:14.757490+00:00","updated_at":"2026-07-05T10:28:14.757490+00:00"}