{"paper":{"title":"SQ Lower Bounds for Learning Bounded Covariance GMMs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.ST","stat.ML","stat.TH"],"primary_cat":"cs.LG","authors_text":"Daniel M. Kane, Ilias Diakonikolas, Nikos Zarifis, Thanasis Pittas","submitted_at":"2023-06-22T17:23:36Z","abstract_excerpt":"We study the complexity of learning mixtures of separated Gaussians with common unknown bounded covariance matrix. Specifically, we focus on learning Gaussian mixture models (GMMs) on $\\mathbb{R}^d$ of the form $P= \\sum_{i=1}^k w_i \\mathcal{N}(\\boldsymbol \\mu_i,\\mathbf \\Sigma_i)$, where $\\mathbf \\Sigma_i = \\mathbf \\Sigma \\preceq \\mathbf I$ and $\\min_{i \\neq j} \\| \\boldsymbol \\mu_i - \\boldsymbol \\mu_j\\|_2 \\geq k^\\epsilon$ for some $\\epsilon>0$. Known learning algorithms for this family of GMMs have complexity $(dk)^{O(1/\\epsilon)}$. In this work, we prove that any Statistical Query (SQ) algorit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2306.13057","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/2306.13057/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"}