Efficient High-Dimensional Approximation: ANOVA Decomposition Meets Wavelets and Random Fourier Features

Abstract

In this thesis, we focus on the problem of reconstructing a multivariate function from discrete d-dimensional samples. Beyond achieving accurate function recovery, we aim to enhance interpretability by identifying how individual variables and their interactions influence the target function. To this end, we develop several efficient hybrid methods that combine the ANOVA decomposition, wavelet techniques, and random Fourier features. The multi-resolution capabilities of wavelets and the scalability of random Fourier features, paired with the interpretability provided by the ANOVA decomposition, enable a robust framework for high-dimensional function approximation. The approaches in this thesis address both computational efficiency and transparency. The total approximation error is influenced by three main components. First, the ANOVA truncation to a function of low effective dimension is the basis for the construction of ANOVA-boosting algorithms, which exploit the structure of the function. Second, the projection onto a finite-dimensional subspace is determined by the choice of basis functions. To analyze the projection error, we explore and discuss wavelet characterizations of functions in certain function spaces, like Sobolev and Besov spaces. Finally, for the regression from samples, we give error bounds for the least squares approximation, which asymptotically coincides with the behavior of the projection error.:1 Introduction 2 Toolbox for function approximation 3 The ANOVA decomposition and the curse of dimensionality 4 Characterizations in function spaces 5 Function approximation from samples 6 Numerical results and application