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 linear
feature learning with non-parametric prediction, which simultaneously estimates
the prediction function and the linear subspace. 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 utilising
alternative minimisation, we iteratively rotate the data to improve alignment
with leading directions and accurately estimate the relevant dimension in
practical settings. We establish that our method yields a consistent estimator
of the prediction function with explicit rates. Additionally, we provide
empirical results demonstrating the performance of RegFeaL in various
experiments.Comment: 42 pages, 5 figure