Automatic locally adaptive smoothing for tree-based set estimation

Tree-based methods similar to CART have recently been utilized for problems in which the main goal is to estimate some set of interest. It is often the case that the boundary of the true set is smooth in some sense, however tree-based estimates will not be smooth, as they will be a union of ‘boxes’. We propose a general methodology for smoothing such sets that allows for varying levels of smoothness on the boundary automatically. The method is similar to the idea underlying support vector machines, which is applying a computationally simple technique to data after a non-linear mapping to produce smooth estimates in the original space. In particular, we consider the problem of level-set estimation for regression functions and the dyadic tree-based method of Willett and Nowak Minimax optimal level-set estimation, IEEE Trans. Image Process. 16 (2007), pp. 2965–2979].