{"paper":{"title":"Costs of Arbitrary Real Matrix Factorizations for Pure-DP Continual Counting","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DS","math.FA"],"primary_cat":"cs.CR","authors_text":"Awnon Bhowmik, Mahmudul Hasan","submitted_at":"2026-07-30T14:41:34Z","abstract_excerpt":"Let \\(T_n\\) be the lower-triangular prefix-sum matrix and let \\(\\cfrob(T_n)\\) and \\(\\ctwo(T_n)\\) be the factorization costs that govern mean and maximum per-coordinate squared error of the Laplace matrix mechanism under pure \\(\\eps\\)-differential privacy, for \\(\\eps>0\\). We prove \\(\\cfrob(T_n),\\ctwo(T_n)=\\Theta\\bigl((\\log(n+1))^{3/2}\\bigr)\\) with no sign, sparsity, or squareness restriction and with arbitrary finite inner dimension. Consequently, within the pure-\\(\\eps\\)-DP matrix-mechanism class the optimized maximum and mean squared errors are both \\(\\Theta(\\eps^{-2}\\log^{3}(n+1))\\). Under t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28703","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.28703/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":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}