Nonparametric Linear Feature Learning in Regression Through Regularisation
This work addresses the challenge of automated feature selection in high-dimensional data for supervised learning, which is incremental as it builds on existing non-parametric methods by incorporating regularization and iterative alignment.
The paper tackles the problem of learning lower-dimensional linear features in high-dimensional regression, where the true information lies in a multi-index model, by proposing a novel method for joint linear feature learning and non-parametric function estimation. The result shows that the expected risk of their method converges to the minimal risk with explicit rates, and empirical results demonstrate its performance in experiments.
Representation learning plays a crucial role in automated feature selection, particularly in the context of high-dimensional data, where non-parametric methods often struggle. In this study, we focus on supervised learning scenarios where the pertinent information resides within a lower-dimensional linear subspace of the data, namely the multi-index model. If this subspace were known, it would greatly enhance prediction, computation, and interpretation. To address this challenge, we propose a novel method for joint linear feature learning and non-parametric function estimation, aimed at more effectively leveraging hidden features for learning. Our approach employs empirical risk minimisation, augmented with a penalty on function derivatives, ensuring versatility. Leveraging the orthogonality and rotation invariance properties of Hermite polynomials, we introduce our estimator, named RegFeaL. By using alternative minimisation, we iteratively rotate the data to improve alignment with leading directions. We establish that the expected risk of our method converges in high-probability to the minimal risk under minimal assumptions and with explicit rates. Additionally, we provide empirical results demonstrating the performance of RegFeaL in various experiments.