{"paper":{"title":"Private Geometric Median in Nearly-Linear Time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CR","cs.LG","stat.ML"],"primary_cat":"cs.DS","authors_text":"Chutong Yang, Daogao Liu, Kevin Tian, Syamantak Kumar","submitted_at":"2025-05-26T16:32:49Z","abstract_excerpt":"Estimating the geometric median of a dataset is a robust counterpart to mean estimation, and is a fundamental problem in computational geometry. Recently, [HSU24] gave an $(\\varepsilon, \\delta)$-differentially private algorithm obtaining an $\\alpha$-multiplicative approximation to the geometric median objective, $\\frac 1 n \\sum_{i \\in [n]} \\|\\cdot - \\mathbf{x}_i\\|$, given a dataset $\\mathcal{D} := \\{\\mathbf{x}_i\\}_{i \\in [n]} \\subset \\mathbb{R}^d$. Their algorithm requires $n \\gtrsim \\sqrt d \\cdot \\frac 1 {\\alpha\\varepsilon}$ samples, which they prove is information-theoretically optimal. This"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.20189","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/2505.20189/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"}