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Let $A_1,\\dots,A_m$ be positive semidefinite \\(d\\times d\\) matrices, and let $\\lambda_1,\\dots,\\lambda_m \\ge 0$ satisfy \\[ \\sum_{i=1}^m \\lambda_i = 1, \\qquad \\sum_{i=1}^m \\lambda_i A_i = I_d, \\qquad \\|A_i\\| \\le M \\quad\\text{for all } i=1,\\dots,m. \\] We show that there exists a deterministic sequence of indices $i_1,i_2,\\dots \\in \\{1,\\dots,m\\}$ such that for every integer $k \\ge 1$, \\[ \\left\\| \\frac{1}{k}\\sum_{r=1}^k A_{i_r} - I_d \\right\\| \\le \\begin{cases} \\displaystyle \\frac{2M\\ln(2d)}{"},"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":true},"canonical_record":{"source":{"id":"2604.06439","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-04-07T20:28:22Z","cross_cats_sorted":[],"title_canon_sha256":"de3a4a85f7c05e6796f29681e7986944ee950ad82e5334c7eb9084acb25c241d","abstract_canon_sha256":"4551d89704857983af055547baa8a0bd36826494c69a5cfaf8d1ab6d28c9db77"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:04:01.628213Z","signature_b64":"Q/9tn29w2OwhvcuQGcT7ZrtqF0R8PBU6OKLIyOCnJBj0JYmBTXIst8/fN8xxf0mwrSdwPK2bKHAxg6uWg/tLBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5fea28b24a4095d1e9f44bcda51c0fd7595ed6c1668d57560b66d4a6f4882e76","last_reissued_at":"2026-05-22T01:04:01.627509Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:04:01.627509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Greedy sparsifications of sums of positive semidefinite matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"There exists a deterministic sequence of positive semidefinite matrices whose partial averages converge to the identity with explicit error bounds in operator norm.","cross_cats":[],"primary_cat":"math.FA","authors_text":"Grigory Ivanov","submitted_at":"2026-04-07T20:28:22Z","abstract_excerpt":"We prove a deterministic analogue of Rudelson's sampling theorem for sums of positive semidefinite matrices. Let $A_1,\\dots,A_m$ be positive semidefinite \\(d\\times d\\) matrices, and let $\\lambda_1,\\dots,\\lambda_m \\ge 0$ satisfy \\[ \\sum_{i=1}^m \\lambda_i = 1, \\qquad \\sum_{i=1}^m \\lambda_i A_i = I_d, \\qquad \\|A_i\\| \\le M \\quad\\text{for all } i=1,\\dots,m. \\] We show that there exists a deterministic sequence of indices $i_1,i_2,\\dots \\in \\{1,\\dots,m\\}$ such that for every integer $k \\ge 1$, \\[ \\left\\| \\frac{1}{k}\\sum_{r=1}^k A_{i_r} - I_d \\right\\| \\le \\begin{cases} \\displaystyle \\frac{2M\\ln(2d)}{"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that there exists a deterministic sequence of indices i1,i2,… such that for every integer k≥1, ||(1/k) sum_{r=1}^k A_{ir} - I_d|| ≤ 2M ln(2d)/k if k ≤ M ln(2d), and ≤ 3 sqrt(M ln(2d)/k) otherwise.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The result assumes there exist lambda_i ≥0 summing to 1 with sum lambda_i A_i = I_d and ||A_i||≤M for all i; if no such convex combination exists, the theorem gives no information. The proof must also construct or guarantee the sequence under only these hypotheses.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"There exists a deterministic sequence of the given PSD matrices such that the k-term average deviates from the identity by at most 2M ln(2d)/k when k is small and 3 sqrt(M ln(2d)/k) when k is large, enabling epsilon-approximations with N = O(M log d / eps^2) terms.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"There exists a deterministic sequence of positive semidefinite matrices whose partial averages converge to the identity with explicit error bounds in operator norm.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7e6fc82bf66d089e2b3d383478e9d1ae5838cdbb1a0d322674e8cb2b43c8b5f0"},"source":{"id":"2604.06439","kind":"arxiv","version":2},"verdict":{"id":"e11ebe43-e902-4c08-ba18-d245c6777c81","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T17:56:56.515356Z","strongest_claim":"We show that there exists a deterministic sequence of indices i1,i2,… such that for every integer k≥1, ||(1/k) sum_{r=1}^k A_{ir} - I_d|| ≤ 2M ln(2d)/k if k ≤ M ln(2d), and ≤ 3 sqrt(M ln(2d)/k) otherwise.","one_line_summary":"There exists a deterministic sequence of the given PSD matrices such that the k-term average deviates from the identity by at most 2M ln(2d)/k when k is small and 3 sqrt(M ln(2d)/k) when k is large, enabling epsilon-approximations with N = O(M log d / eps^2) terms.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The result assumes there exist lambda_i ≥0 summing to 1 with sum lambda_i A_i = I_d and ||A_i||≤M for all i; if no such convex combination exists, the theorem gives no information. 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