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.

翻译：我们提出了快速切比雪夫变换（FCT），这是一种基于非零切比雪夫系数位置信息的快速随机算法，用于计算高维函数的切比雪夫逼近。该算法并非在每个维度上采样全分辨率切比雪夫网格，而是随机采样多个不同分辨率的网格，并在系数空间中求解最小二乘问题，从而同时跨所有网格计算逼近目标函数的多项式。我们从理论和实证两方面证明，FCT在复杂高维问题上展现出拟线性缩放特性和高数值精度。我们通过与替代切比雪夫逼近方案的对比，验证了该方法的有效性。特别地，我们强调了算法在高维场景下的卓越表现，其相较于常用替代技术实现了显著的加速效果。