Joseph Meyer

Abstract: Estimation in regression and classification problems which include low dimensional structures are considered. The underlying question is the following. How well do statistical learning methods perform for models with low dimensional structures? We approach this question using various algorithms in various settings. First, we introduce a new tree based algorithm we named random planted forest. It adapts particularly well to models which consist of low dimensional structures. We examine its performance in simulation studies and include some theoretical backing by proving optimal convergence rates in certain settings for a modification of the algorithm. A generalized version of the algorithm is included, which can be used in classification settings. Furthermore, we prove optimal convergence rates in a classification setting using neural networks. While non-optimal rates existed for this problem, we are the first to prove optimal ones.