In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to accomplish the tensor compression, provide a detailed analysis of the computational costs, provide insight into the error of the resulting approximations, and discuss the benefits of the proposed approaches. We also apply the tensor-based function approximation to develop low-rank matrix approximations to kernel matrices that describe pairwise interactions between two sets of points; the resulting low-rank approximations are efficient to compute and store (the complexity is linear in the number of points). We have detailed numerical experiments on example problems involving multivariate function approximation, low-rank matrix approximations of kernel matrices involving well-separated clusters of sources and target points, and a global low-rank approximation of kernel matrices with an application to Gaussian processes.
翻译:在本文中,我们采用多种变量函数近似的方法,使用功能评价、Chebyshev 多元分子和以调压为基础的压缩技术使用塔克格式。我们开发了新颖的随机技术,以完成高压压缩,对计算成本进行详细分析,对由此产生的近似错误进行深入分析,并讨论拟议方法的效益。我们还采用基于 ARO 的函数近近似法,对内核矩阵进行低级矩阵近似,以描述两组点之间的对称互动;由此产生的低级近似法对计算和存储有效(点数的复杂度为线性 ) 。 我们对涉及多变量函数近似、涉及源和目标点的低端内核矩阵近似值,以及包含高斯进程应用程序的内核质矩阵全球低端近近似值等问题进行了详细的数字实验。