{"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. The proof must also construct or guarantee the sequence under only these hypotheses.","pith_extraction_headline":"There exists a deterministic sequence of positive semidefinite matrices whose partial averages converge to the identity with explicit error bounds in operator norm."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.06439/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"17411bbec0eab82c2f3ca5497c0545128c05dec28a708869c55ff710a3b06e3a"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}