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Previously no efficient algorithm can decompose 3rd order tensors when the rank is super-linear in the dimension. Using ideas from sum-of-squares hierarchy, we give the first quasi-polynomial time algorithm that can decompose a random 3rd order tensor decomposition when the rank is as large as $n^{3/2}/\\textrm{polylog} n$.\n  We also give a"},"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":"1504.05287","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-04-21T03:21:53Z","cross_cats_sorted":["cs.LG","stat.ML"],"title_canon_sha256":"6caba22ac7269fbf3f7c40f953e7c169bed0ef344192230118515835f328b6ef","abstract_canon_sha256":"2197a95014cbcd2ec569f2292bb4c995898b91db8fa16726fa6beb06cdf5d9f4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:13.191420Z","signature_b64":"sZd0nLNOzZqdfSzRjBak8HpRo3OrKxN2nTkcYgXO3ArdoOE3f0F/vh+TIt5L8TRAL2lHIlPy3aHyh2JMar3UAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4cbca21e45c24118992aeb7c6eefd867884e46c7afc56bf82c4a200d14abbc1d","last_reissued_at":"2026-05-18T02:18:13.190895Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:13.190895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decomposing Overcomplete 3rd Order Tensors using Sum-of-Squares Algorithms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","stat.ML"],"primary_cat":"cs.DS","authors_text":"Rong Ge, Tengyu Ma","submitted_at":"2015-04-21T03:21:53Z","abstract_excerpt":"Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. 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