Nonparametric Linear Feature Learning in Regression Through Regularisation

TL;DR

This paper introduces RegFeaL, a novel method for joint linear feature learning and non-parametric function estimation in high-dimensional regression, leveraging Hermite polynomials and iterative data rotation to improve prediction and interpretability.

Contribution

It proposes a new estimator combining empirical risk minimisation with derivative penalties, with theoretical guarantees and empirical validation for high-dimensional feature learning.

Findings

Converges to minimal risk with explicit rates

Effective in high-dimensional settings

Demonstrates strong empirical performance

Abstract

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.