通过正交多项式展开的量化张量列中函数的逼近与复合
Approximation and composition of functions in quantized tensor trains via orthogonal polynomial expansions
AI总结：
本文提出一种基于正交多项式展开的构造性算法，将单变量和多变量函数表示为量化张量列（QTT）逼近器，支持函数复合，在切比雪夫基下快速收敛且数值稳定，性能优于或媲美TCI等方法。
摘要：
这项工作探讨了单变量和多变量函数在量化张量列（QTT）中的表示。它开发了一种构造性算法，利用正交多项式基的展开和Clenshaw求值，将解析函数和高可微函数表示为QTT逼近器。该方法为QTT形式的函数复合提供了一个通用框架，并能高效推广到多维设置。在不失一般性的前提下，本文聚焦于切比雪夫基，在该基下，该方法对高可微函数展现出快速收敛，与经典理论预测一致，并在高多项式阶数下保持数值稳定。该算法的性能与张量交叉插值（TCI）和多尺度插值构造进行了比较，在特定场景下表现出有竞争力的性能和良好的扩展性。
英文摘要：
This work explores the representation of univariate and multivariate functions as quantized tensor trains (QTT). It develops a constructive algorithm that employs expansions in orthogonal polynomial bases and Clenshaw evaluations to represent analytic and highly differentiable functions as QTT approximants. This approach provides a general framework for function composition in QTT form that generalizes efficiently to multidimensional settings. Without loss of generality, the presentation focuses on the Chebyshev basis, where the method demonstrates rapid convergence for highly differentiable functions, in agreement with classical theoretical predictions, and remains numerically stable at high polynomial orders. The performance of the algorithm is compared with that of tensor cross-interpolation (TCI) and multiscale interpolative constructions, demonstrating competitive performance and favorable scaling behavior in certain scenarios.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
