Tensor train construction from tensor actions, with application to compression of large high order derivative tensors
Tensor train construction from tensor actions, with application to compression of large high order derivative tensors
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DOI：
10.1137/20m131936x
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发表时间：
2020-01
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通讯作者：
Nick Alger;Peng Chen;O. Ghattas
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作者：
Nick Alger;Peng Chen;O. Ghattas
我们提出了一种基于张量作为向量值多线性函数的动作将张量转换为张量序列格式的方法。构建张量序列的现有方法需要访问张量的“数组条目”，因此如果张量只能通过其操作来访问，则效率低下或计算量过大，尤其是对于高阶张量。我们的方法允许对通过方程组的解隐式定义的非线性映射的大型高阶导数张量进行有效的张量序列压缩。这些导数张量的数组条目无法直接访问，但可以通过我们讨论的过程有效地计算这些张量的动作。理论上，此类张量通常适合张量序列压缩，但到目前为止，还没有有效的算法将它们转换为张量序列格式。我们通过压缩大小为 $41 \times 42 \times 43 \times 44 \times 45$ 的希尔伯特张量，并通过形成具有边界输出的随机偏微分方程的噪声白化参数到输出映射的高阶（高达 $5^\text{th}$ 阶导数/$6^\text{th}$ 阶张量）泰勒级数代理来演示我们的方法。
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.