An algorithmic Framework for Multiresolution based non-parametric Regression
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We study a method to solve non-parametric regression problems in one and two dimensions with statistical multiresolution estimation. We present the non-parametric regression problem, then introduce the multiresolution norm and use it to formulate the optimization problem. We will discuss two different regularization terms, a quadratic regularization term and a total variation term. We will solve the quadratic problem in both one and two dimensions. In order the solve this problem in one dimension, we use the ADMM (alternating direction method of multipliers) and Dykstra's projection method. For the two dimensional case, we use the Douglas-Rachford method. We will consider the total variation problem only in two dimensions. To solve the total variation problem we use the Douglas-Rachford method and Chambolle's projection method. Towards the end, we will verify and test the algorithms numerically.