{"paper":{"title":"Phase transition and higher order analysis of $L_q$ regularization under dependence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.CO","stat.TH"],"primary_cat":"math.ST","authors_text":"Hanwen Huang, Peng Zeng, Qinglong Yang","submitted_at":"2022-11-19T00:02:01Z","abstract_excerpt":"We study the problem of estimating a $k$-sparse signal ${\\mbox{$\\beta$}}_0\\in{\\bf R}^p$ from a set of noisy observations ${\\bf y}\\in{\\bf R}^n$ under the model ${\\bf y}={\\bf X}{\\mbox{$\\beta$}}+{\\bf w}$, where ${\\bf X}\\in{\\bf R}^{n\\times p}$ is the measurement matrix the row of which is drawn from distribution $N(0,{\\mbox{$\\Sigma$}})$. We consider the class of $L_q$-regularized least squares (LQLS) given by the formulation $\\hat{\\mbox{$\\beta$}}(\\lambda,q)=\\text{argmin}_{{\\mbox{$\\beta$}}\\in{\\bf R}^p}\\frac{1}{2}\\|{\\bf y}-{\\bf X}{\\mbox{$\\beta$}}\\|^2_2+\\lambda\\|{\\mbox{$\\beta$}}\\|_q^q$, where $\\|\\cdo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.10541","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2211.10541/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"}