A Sparse Fast Chebyshev Transform for High-Dimensional Approximation

TL;DR

This paper introduces a randomized, fast algorithm for high-dimensional Chebyshev approximation that significantly reduces computational complexity while maintaining high accuracy, outperforming existing methods.

Contribution

The paper presents the Fast Chebyshev Transform (FCT), a novel randomized algorithm that efficiently approximates functions in high dimensions using sparse sampling and least-squares fitting.

Findings

FCT exhibits quasi-linear scaling with problem size.

FCT achieves high numerical accuracy on complex high-dimensional problems.

FCT provides significant speedups over traditional Chebyshev approximation methods.

Abstract

We present the Fast Chebyshev Transform (FCT), a fast, randomized algorithm to compute a Chebyshev approximation of functions in high-dimensions from the knowledge of the location of its nonzero Chebyshev coefficients. Rather than sampling a full-resolution Chebyshev grid in each dimension, we randomly sample several grids with varied resolutions and solve a least-squares problem in coefficient space in order to compute a polynomial approximating the function of interest across all grids simultaneously. We theoretically and empirically show that the FCT exhibits quasi-linear scaling and high numerical accuracy on challenging and complex high-dimensional problems. We demonstrate the effectiveness of our approach compared to alternative Chebyshev approximation schemes. In particular, we highlight our algorithm's effectiveness in high dimensions, demonstrating significant speedups over commonly-used alternative techniques.