Adaptation in Shape-Constrained Regression Problems

• Activity Number: 218
• Type: Invited
• Date/Time: Monday, August 10, 2015 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Bodhisattva Sen* and Adityanand Guntuboyina
• Companies: Columbia University and UC Berkeley
• Keywords: isotonic regression ; global risk bounds ; convex regression ; projection

Abstract

We consider the problem of estimating a normal mean constrained to be in a convex polyhedral cone in an Euclidean space. We say that the true mean is sparse if it belongs to a low dimensional face of the cone. We show that, in a certain natural subclass of these problems, the maximum likelihood estimator automatically adapts to sparsity in the underlying true mean. We illustrate this phenomenon by deriving risk (upper) bounds for the estimator. We discuss the problems of convex regression and univariate and bivariate isotonic regression as examples.