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Consider the problem of estimating the mean of a Gaussian random vector whenthe mean vector is assumed to be in a given convex set. The most naturalsolution is to take the Euclidean projection of the data vector on to thisconvex set; in other words, performing "least squares under a convexconstraint." Many problems in modern statistics and statistical signalprocessing theory are special cases of this general situation. Examples includethe lasso and other high-dimensional regression techniques, function estimationproblems, matrix estimation and completion, shape-restricted regression,constrained denoising, linear inverse problems, etc. This paper presents threegeneral results about this problem, namely, (a) an exact computation of themain term in the estimation error by relating it to expected maxima of Gaussianprocesses (existing results only give upper bounds), (b) a theorem showing thatthe least squares estimator is always admissible up to a universal constant inany problem of the above kind and (c) a counterexample showing that leastsquares 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 twoexamples of contemporary interest.

A new perspective on least squares under convex constraint