论文标题
张量的张量火车构造从张量操作中，并用于压缩大型衍生量张量
Tensor train construction from tensor actions, with application to compression of large high order derivative tensors
论文作者
论文摘要
我们提出了一种基于张量作为矢量值值的多线性函数的张量的操作，将张量转换为张量列车格式。现有的构造张量列车的方法需要访问张量的“数组条目”，因此，只有通过其操作即可访问张量，尤其是对于高阶张量，效率低下或计算效率不高。我们的方法允许对非线性映射的大型高阶衍生量张量进行有效的张量列车压缩，这些映射是通过方程式解决方案隐式定义的。这些导数张量的数组条目无法直接访问，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，这种张量通常适合张量列车的压缩，但是直到现在，还没有有效的算法将它们转换为张量火车的格式。我们通过压缩$ 41 \ times 42 \ times 43 \ times 44 \ times 45 $的希尔伯特张量来演示我们的方法边界输出。
We present a method for converting tensors into 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 $41 \times 42 \times 43 \times 44 \times 45$, and by forming high order (up to $5^\text{th}$ order derivatives/$6^\text{th}$ order tensors) Taylor series surrogates of the noise-whitened parameter-to-output map for a stochastic partial differential equation with boundary output.
