{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,25]],"date-time":"2026-10-25T14:28:27Z","timestamp":1792938507410,"version":"4.5.0"},"reference-count":82,"publisher":"Institute of Mathematical Statistics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Statist. Sci."],"published-print":{"date-parts":[[2012,11,1]]},"abstract":"<jats:p>High-dimensional statistical inference deals with models in which the the number of parameters $p$ is comparable to or larger than the sample size $n$. Since it is usually impossible to obtain consistent procedures unless $p\/n\\rightarrow0$, a line of recent work has studied models with various types of low-dimensional structure, including sparse vectors, sparse and structured matrices, low-rank matrices and combinations thereof. In such settings, a general approach to estimation is to solve a regularized optimization problem, which combines a loss function measuring how well the model fits the data with some regularization function that encourages the assumed structure. This paper provides a unified framework for establishing consistency and convergence rates for such regularized $M$-estimators under high-dimensional scaling. We state one main theorem and show how it can be used to re-derive some existing results, and also to obtain a number of new results on consistency and convergence rates, in both $\\ell_{2}$-error and related norms. Our analysis also identifies two key properties of loss and regularization functions, referred to as restricted strong convexity and decomposability, that ensure corresponding regularized $M$-estimators have fast convergence rates and which are optimal in many well-studied cases.<\/jats:p>","DOI":"10.1214\/12-sts400","type":"journal-article","created":{"date-parts":[[2012,12,21]],"date-time":"2012-12-21T09:03:21Z","timestamp":1356080601000},"page":"538-557","source":"Crossref","is-referenced-by-count":584,"title":["A Unified Framework for High-Dimensional Analysis of $M$-Estimators with Decomposable Regularizers"],"prefix":"10.1214","volume":"27","author":[{"given":"Sahand N.","family":"Negahban","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pradeep","family":"Ravikumar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Martin J.","family":"Wainwright","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bin","family":"Yu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"15","doi-asserted-by":"publisher","unstructured":"[15] Cand\u00e8s, E. 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