标题： 关于凸约束下的最小二乘法的新视角 显示英文标题

标题： A new perspective on least squares under convex constraint

摘要： 考虑估计高斯随机向量均值的问题，其中假设均值向量位于某个凸集内。 最自然的解法是将数据向量在欧几里得空间中投影到该凸集上；换句话说，执行“在凸约束下的最小二乘法”。 现代统计学和统计信号处理理论中的许多问题都是这一一般情形的特例。 举例来说，包括套索（lasso）和其他高维回归技术、函数估计问题、矩阵估计与补全、形状受限回归、约束去噪、线性逆问题等。 本文提出了关于此问题的三个通用结果，即：(a) 通过将其与高斯过程的最大期望联系起来，精确计算估计误差的主要项（现有结果仅提供上界），(b) 一个定理表明，对于上述任何一类问题，最小二乘估计量总是以一个普适常数为界的可接受估计量，以及 (c) 一个反例表明最小二乘估计量并不总是达到渐近最优率。 接着，(a) 中的结果被用于计算两个当代感兴趣的例子中最小二乘估计器的误差。 显示英文摘要

摘要： 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.