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：我们提出快速切比雪夫变换（FCT），这是一种快速、随机化的算法，可利用非零切比雪夫系数的位置信息，计算高维函数的切比雪夫逼近。该方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，从而计算出同时逼近所有网格上目标函数的多项式。我们从理论和实验上证明，FCT在具有挑战性的复杂高维问题中表现出拟线性扩展性和高数值精度。我们展示了所提方法相较于其他切比雪夫逼近方案的有效性，尤其强调了算法在高维问题上的效力，与常用替代技术相比实现了显著的加速。