{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:I7IU5O7RVICSFSAAOFRTLE4QH5","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":"3e96e377b3f8895e5c2a3e4cd748add174a4e95ec9fcce13676b484b04047613","cross_cats_sorted":["cs.LG","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-10-05T18:09:50Z","title_canon_sha256":"dce0c66c750e23e4194e0066d597839dc7a218f6048d626ea33ba4bb22107e66"},"schema_version":"1.0","source":{"id":"1310.1502","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.1502","created_at":"2026-05-18T02:51:49Z"},{"alias_kind":"arxiv_version","alias_value":"1310.1502v3","created_at":"2026-05-18T02:51:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.1502","created_at":"2026-05-18T02:51:49Z"},{"alias_kind":"pith_short_12","alias_value":"I7IU5O7RVICS","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"I7IU5O7RVICSFSAA","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"I7IU5O7R","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:cb4f5387f97ecd862091e509f73417ffca8a803b7b3f22fbde65d9797da1607f","target":"graph","created_at":"2026-05-18T02:51:49Z","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":"Given a real matrix A with n columns, the problem is to approximate the Gram product AA^T by c << n weighted outer products of columns of A. Necessary and sufficient conditions for the exact computation of AA^T (in exact arithmetic) from c >= rank(A) columns depend on the right singular vector matrix of A. For a Monte-Carlo matrix multiplication algorithm by Drineas et al. that samples outer products, we present probabilistic bounds for the 2-norm relative error due to randomization. The bounds depend on the stable rank or the rank of A, but not on the matrix dimensions. Numerical experiments ","authors_text":"Ilse C. F. Ipsen, John T. Holodnak","cross_cats":["cs.LG","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-10-05T18:09:50Z","title":"Randomized Approximation of the Gram Matrix: Exact Computation and Probabilistic Bounds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.1502","kind":"arxiv","version":3},"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":"sha256:0ef5ab87df09adb81b2d1c5f3983b61810db1c8226c4fb53dda77dff7c607530","target":"record","created_at":"2026-05-18T02:51:49Z","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":"3e96e377b3f8895e5c2a3e4cd748add174a4e95ec9fcce13676b484b04047613","cross_cats_sorted":["cs.LG","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2013-10-05T18:09:50Z","title_canon_sha256":"dce0c66c750e23e4194e0066d597839dc7a218f6048d626ea33ba4bb22107e66"},"schema_version":"1.0","source":{"id":"1310.1502","kind":"arxiv","version":3}},"canonical_sha256":"47d14ebbf1aa0522c80071633593903f6c48ec0d13a905888fafebca56bd6b1e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47d14ebbf1aa0522c80071633593903f6c48ec0d13a905888fafebca56bd6b1e","first_computed_at":"2026-05-18T02:51:49.579604Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:51:49.579604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BXzHr7sVQHvEAhE/CB2OYHz/OwkV8CvK3KxEu3t2TXpjJIaX6GhsMSEOa9r1MUESSk5dhx4SmgMxOBJexnjcCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:51:49.580087Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.1502","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0ef5ab87df09adb81b2d1c5f3983b61810db1c8226c4fb53dda77dff7c607530","sha256:cb4f5387f97ecd862091e509f73417ffca8a803b7b3f22fbde65d9797da1607f"],"state_sha256":"801f1a166f885edd8948e1066c59f7ce56bf9115a8c9cc9485ebca001ed64538"}