Nonparametric sparsity and regularization

Abstract

In this work we are interested in the problems of supervised learning and variable selection when the input-output dependence is described by a nonlinear function depending on a few variables. Our goal is to consider a sparse nonparametric model, hence avoiding linear or additive models. The key idea is to measure the importance of each variable in the model by making use of partial derivatives. Based on this intuition we propose a new notion of nonparametric sparsity and a corresponding least squares regularization scheme. Using concepts and results from the theory of reproducing kernel Hilbert spaces and proximal methods, we show that the proposed learning algorithm corresponds to a minimization problem which can be provably solved by an iterative procedure. The consistency properties of the obtained estimator are studied both in terms of prediction and selection performance. An extensive empirical analysis shows that the proposed method performs favorably with respect to the state-of-the-art methods.

45 pages, 11 figures

References in corpus (6)

• Consistency of the group Lasso and multiple kernel learning
• The composite absolute penalties family for grouped and hierarchical variable selection
• Component selection and smoothing in multivariate nonparametric regression
• Sparsity in multiple kernel learning
• High-Dimensional Non-Linear Variable Selection through Hierarchical Kernel Learning
• Selection of variables and dimension reduction in high-dimensional non-parametric regression