Adaptation via convex optimization in two nonparametric estimation problems
Seminar | September 20 | 4-5 p.m. | 1011 Evans Hall
Adityanand Guntuboyina, University of California, Berkeley
We study two convex optimization based procedures for nonparametric function estimation: trend filtering (or higher order total variation denoising) and the Kiefer-Wolfowitz MLE for Gaussian location mixtures. Trend filtering can be seen as a technique for fitting spline-like functions for nonparametric regression with adaptive knot selection. It can also be seen as a special case of LASSO for a specific design matrix with highly correlated columns. The Kiefer-Wolfowitz MLE is a technique for nonparametric density estimation with an adaptive selection of the number of Gaussian mixture components. We shall prove that these two procedures have natural adaptive risk behavior for prediction (i.e., they achieve prediction performance that is comparable to appropriate Oracle estimators as well as to non-convex combinatorial procedures) under sparsity-like assumptions. The results for trend filtering are based on joint work with Donovan Lieu, Sabyasachi Chatterjee and Bodhisattva Sen and the results for the Kiefer-Wolfowitz MLE are based on joint work with Sujayam Saha.