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December 2014 A new perspective on least squares under convex constraint
Sourav Chatterjee
Ann. Statist. 42(6): 2340-2381 (December 2014). DOI: 10.1214/14-AOS1254


Consider the problem of estimating the mean of a Gaussian random vector when the mean vector is assumed to be in a given convex set. The most natural solution is to take the Euclidean projection of the data vector on to this convex set; in other words, performing “least squares under a convex constraint.” Many problems in modern statistics and statistical signal processing theory are special cases of this general situation. Examples include the lasso and other high-dimensional regression techniques, function estimation problems, matrix estimation and completion, shape-restricted regression, constrained denoising, linear inverse problems, etc. This paper presents three general results about this problem, namely, (a) an exact computation of the main term in the estimation error by relating it to expected maxima of Gaussian processes (existing results only give upper bounds), (b) a theorem showing that the least squares estimator is always admissible up to a universal constant in any problem of the above kind and (c) a counterexample showing that least squares estimator may not always be minimax rate-optimal. The result from part (a) is then used to compute the error of the least squares estimator in two examples of contemporary interest.


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Sourav Chatterjee. "A new perspective on least squares under convex constraint." Ann. Statist. 42 (6) 2340 - 2381, December 2014.


Published: December 2014
First available in Project Euclid: 20 October 2014

zbMATH: 1302.62053
MathSciNet: MR3269982
Digital Object Identifier: 10.1214/14-AOS1254

Primary: 62F10 , 62F12 , 62F30 , 62G08

Keywords: convex constraint , Denoising , empirical process , isotonic regression , Lasso , least squares , maximum likelihood

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 6 • December 2014
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