High-Dimensional Non-Linear Variable Selection through Hierarchical Kernel Learning

TL;DR

This paper introduces a hierarchical kernel learning method for high-dimensional non-linear variable selection, enabling efficient kernel selection and demonstrating state-of-the-art predictive performance in synthetic and real datasets.

Contribution

It extends multiple kernel learning to a hierarchical structure, allowing polynomial-time kernel selection and high-dimensional variable selection consistency.

Findings

Achieves state-of-the-art predictive accuracy on synthetic datasets.

Performs efficient kernel selection using a graph-adapted sparsity norm.

Supports variable selection with exponential relevance in high-dimensional settings.

Abstract

We consider the problem of high-dimensional non-linear variable selection for supervised learning. Our approach is based on performing linear selection among exponentially many appropriately defined positive definite kernels that characterize non-linear interactions between the original variables. To select efficiently from these many kernels, we use the natural hierarchical structure of the problem to extend the multiple kernel learning framework to kernels that can be embedded in a directed acyclic graph; we show that it is then possible to perform kernel selection through a graph-adapted sparsity-inducing norm, in polynomial time in the number of selected kernels. Moreover, we study the consistency of variable selection in high-dimensional settings, showing that under certain assumptions, our regularization framework allows a number of irrelevant variables which is exponential in the number of observations. Our simulations on synthetic datasets and datasets from the UCI repository show state-of-the-art predictive performance for non-linear regression problems.