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A continuous analogue of the tensor-train decomposition

机译：张量应变分解的连续模拟

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摘要

We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents functions using a tensor-train ansatz by replacing the three-dimensional TT cores with univariate matrix-valued functions. The main contribution of this paper is a framework to compute the FT that employs adaptive approximations of univariate fibers, and that is not tied to any tensorized discretization. The algorithm can be coupled with any univariate linear or nonlinear approximation procedure. We demonstrate that this approach can generate multivariate function approximations that are several orders of magnitude more accurate, for the same cost, than those based on the conventional approach of compressing the coefficient tensor of a tensor-product basis. Our approach is in the spirit of other continuous computation packages such as Chebfun, and yields an algorithm which requires the computation of "continuous" matrix factorizations such as the LU and QR decompositions of vector-valued functions. To support these developments, we describe continuous versions of an approximate maximum-volume cross approximation algorithm and of a rounding algorithm that re-approximates an FT by one of lower ranks. We demonstrate that our technique improves accuracy and robustness, compared to TT and quantics-TT approaches with fixed parameterizations, of high-dimensional integration, differentiation, and approximation of functions with local features such as discontinuities and other nonlinearities. (C) 2018 Elsevier B.V. All rights reserved.
机译：我们开发了新的近似算法和数据结构，用于使用函数张量-应变（FT）（张量-应变（TT）分解的连续扩展）来表示和使用多元函数进行计算。 FT通过使用单变量矩阵值函数替换三维TT核，从而使用张量列ansatz表示函数。本文的主要贡献是一个计算FT的框架，该框架采用单变量纤维的自适应逼近，并且与任何张量离散无关。该算法可以与任何单变量线性或非线性逼近过程耦合。我们证明，与基于传统方法压缩张量积基的系数张量的方法相比，该方法可以以相同的成本生成更精确的几个数量级的多元函数逼近。我们的方法本着其他连续计算程序包（例如Chebfun）的精神，并得出了一种算法，该算法要求计算“连续”矩阵因式分解，例如向量值函数的LU和QR分解。为了支持这些发展，我们描述了近似最大体积交叉近似算法和舍入算法的连续版本，该算法将FT按较低等级之一重新近似。我们证明，与具有固定参数化的TT和quantistics-TT方法相比，我们的技术提高了准确性和鲁棒性，这些方法具有高维积分，微分和具有局部特征（例如不连续性和其他非线性）的函数逼近。 （C）2018 Elsevier B.V.保留所有权利。

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