Approximation of Generalized Ridge Functions in High Dimensions
For researchers in high-dimensional approximation, this provides a novel method to handle functions with low-dimensional structure, though the setting is specific and incremental.
The paper introduces algorithms to approximate linear-sleeve functions (functions depending only on distance to an unknown linear subspace) in high dimensions, proving error bounds and showing numerical effectiveness.
This paper studies the approximation of generalized ridge functions, namely of functions which are constant along some submanifolds of $\mathbb{R}^N$. We introduce the notion of linear-sleeve functions, whose function values only depend on the distance to some unknown linear subspace $L$. We propose two effective algorithms to approximate linear-sleeve functions $f(x)=g(\text{dist}(x,L)^2)$, when both the linear subspace $L\subset \mathbb{R}^N$ and the function $g\in C^s[0,1]$ are unknown. We will prove error bounds for both algorithms and provide an extensive numerical comparison of both. We further propose an approach of how to apply these algorithms to capture general sleeve functions, which are constant along some lower dimensional submanifolds.