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These results can be generalized to the rectangular matrix case."},"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":"1606.03931","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-06-13T13:02:31Z","cross_cats_sorted":[],"title_canon_sha256":"e8240ce7d1b7026171422afce5f6db241f95c23ae117c3cf90fafecfc43df8bd","abstract_canon_sha256":"1a3b5df5422481c5fafe027c8459318b1bc3327cfade9fd6025b09e0483c263a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:02.449568Z","signature_b64":"JQmxWgfTKQRhcQP/i2Wf6oe4LWCJQCzPc4WAE+UzQ0onm1xCPZqTMyT/5pulyGyn21k3TxadoIjBeJhbu1CBBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b60b57ae6d7601ef31110f4ef3e91568aed320ba4238afd51f1081b092f1bbb","last_reissued_at":"2026-05-18T01:10:02.448988Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:02.448988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Upper bound for intermediate singular values of random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Feng Wei","submitted_at":"2016-06-13T13:02:31Z","abstract_excerpt":"In this paper, we prove that an $n\\times n$ matrix $A$ with independent centered subgaussian entries satisfies \\[ s_{n+1-l}(A) \\le C_1t \\frac{l}{\\sqrt{n}} \\] with probability at least $1-\\exp(-C_2tl)$. 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