{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:SNFWZGICQNQTF2XQW5SOIYYAIL","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":"4898c99d760d69542d9c7e90fc4b9cc7f4bf89b26774ce3fbc462c197fbc2c5c","cross_cats_sorted":["cs.IT","math.FA","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-09-08T11:05:17Z","title_canon_sha256":"667c298c955125d7789db36485db487a1ffa9dfb702e20b6afa4e1ae67cce601"},"schema_version":"1.0","source":{"id":"1509.02322","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.02322","created_at":"2026-05-18T01:33:41Z"},{"alias_kind":"arxiv_version","alias_value":"1509.02322v1","created_at":"2026-05-18T01:33:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.02322","created_at":"2026-05-18T01:33:41Z"},{"alias_kind":"pith_short_12","alias_value":"SNFWZGICQNQT","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"SNFWZGICQNQTF2XQ","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"SNFWZGIC","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:4f9d33048dbd7dde45b7c9241ac02294b2d307f4ddfe910aa5956b19e0e20db1","target":"graph","created_at":"2026-05-18T01:33:41Z","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":"Let $A$ be a matrix whose columns $X_1,\\dots, X_N$ are independent random vectors in $\\mathbb{R}^n$. Assume that the tails of the 1-dimensional marginals decay as $\\mathbb{P}(|\\langle X_i, a\\rangle|\\geq t)\\leq t^{-p}$ uniformly in $a\\in S^{n-1}$ and $i\\leq N$. Then for $p>4$ we prove that with high probability $A/{\\sqrt{n}}$ has the Restricted Isometry Property (RIP) provided that Euclidean norms $|X_i|$ are concentrated around $\\sqrt{n}$. We also show that the covariance matrix is well approximated by the empirical covariance matrix and establish corresponding quantitative estimates on the ra","authors_text":"Alain Pajor, Alexander E. Litvak, Nicole Tomczak-Jaegermann, Olivier Gu\\'edon","cross_cats":["cs.IT","math.FA","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-09-08T11:05:17Z","title":"On the interval of fluctuation of the singular values of random matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.02322","kind":"arxiv","version":1},"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":"sha256:57e16af953dcd2898fc8f01fe03f8dac5ab26f1fb43bd3aaa663cbf5fd6899e8","target":"record","created_at":"2026-05-18T01:33:41Z","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":"4898c99d760d69542d9c7e90fc4b9cc7f4bf89b26774ce3fbc462c197fbc2c5c","cross_cats_sorted":["cs.IT","math.FA","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-09-08T11:05:17Z","title_canon_sha256":"667c298c955125d7789db36485db487a1ffa9dfb702e20b6afa4e1ae67cce601"},"schema_version":"1.0","source":{"id":"1509.02322","kind":"arxiv","version":1}},"canonical_sha256":"934b6c9902836132eaf0b764e4630042df4a83292226bd8b24dbb69156c21aba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"934b6c9902836132eaf0b764e4630042df4a83292226bd8b24dbb69156c21aba","first_computed_at":"2026-05-18T01:33:41.219117Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:33:41.219117Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"B5D2huJi2kk6K3ygTpuufNPR3xYmW5B/DIell2VaHDrONXhbXraTjPEN3R6jfVN0sn2XQ4CW6w2lmBQb1iapCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:33:41.219744Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.02322","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57e16af953dcd2898fc8f01fe03f8dac5ab26f1fb43bd3aaa663cbf5fd6899e8","sha256:4f9d33048dbd7dde45b7c9241ac02294b2d307f4ddfe910aa5956b19e0e20db1"],"state_sha256":"7857156a57140ccd3751059b4d8abcec84eb3c93354418c311329dfca087446d"}