SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3516-A3539, January 2020.
We present a method for converting tensors into the tensor train format based on actions of the tensor as a vector-valued multilinear function. Existing methods for constructing tensor trains require access to “array entries” of the tensor and are therefore inefficient or computationally prohibitive if the tensor is accessible only through its action, especially for high order tensors. Our method permits efficient tensor train compression of large high order derivative tensors for nonlinear mappings that are implicitly defined through the solution of a system of equations. Array entries of these derivative tensors are not directly accessible, but actions of these tensors can be computed efficiently via a procedure that we discuss. Such tensors are often amenable to tensor train compression in theory, but until now no efficient algorithm existed to convert them into tensor train format. We demonstrate our method by compressing a Hilbert tensor of size 41 x 42 x 43 x 44 x 45, and by forming high order (up to fifth order derivatives/sixth order tensors) Taylor series surrogates of the noise-whitened parameter-to-output map for a stochastic partial differential equation with boundary output.

中文翻译：

---

从张量动作构造张量列车，并应用于大型高阶导数张量的压缩

SIAM科学计算杂志，第42卷，第5期，第A3516-A3539页，2020年1月。
我们提出了一种基于张量作为矢量值多线性函数将张量转换为张量列格式的方法。现有的构造张量列的方法需要访问张量的“数组项”，因此，如果仅通过张量的操作才能访问张量（特别是对于高阶张量），则效率低下或在计算上受到禁止。对于非线性映射，我们的方法允许对大型高阶导数张量进行有效的张量列压缩，这些非线性映射是通过方程组的解决方案隐式定义的。这些导数张量的数组条目不能直接访问，但是可以通过我们讨论的过程有效地计算这些张量的动作。从理论上讲，此类张量通常适合张量训练压缩，但是到目前为止，还没有有效的算法可以将它们转换为张量序列格式。我们通过压缩大小为41 x 42 x 43 x 44 x 45的希尔伯特张量并形成高阶（至多五阶导数/六阶张量）的泰勒级数代数来证明我们的方法，该代数是白噪声参数输出带边界输出的随机偏微分方程的映射。