{"paper":{"title":"Near-Optimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Log-concave Densities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","cs.LG","math.IT","stat.TH"],"primary_cat":"math.ST","authors_text":"Alistair Stewart, Anastasios Sidiropoulos, Ilias Diakonikolas, Timothy Carpenter","submitted_at":"2018-02-28T18:32:07Z","abstract_excerpt":"We study the problem of learning multivariate log-concave densities with respect to a global loss function. We obtain the first upper bound on the sample complexity of the maximum likelihood estimator (MLE) for a log-concave density on $\\mathbb{R}^d$, for all $d \\geq 4$. Prior to this work, no finite sample upper bound was known for this estimator in more than $3$ dimensions.\n  In more detail, we prove that for any $d \\geq 1$ and $\\epsilon>0$, given $\\tilde{O}_d((1/\\epsilon)^{(d+3)/2})$ samples drawn from an unknown log-concave density $f_0$ on $\\mathbb{R}^d$, the MLE outputs a hypothesis $h$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.10575","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}