Tensor train construction from tensor actions, with application to compression of large high order derivative tensors
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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 easily computable, but their actions 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 practical algorithm existed to convert them into tensor train format. We demonstrate our method by compressing a Hilbert tensor of size 41 × 42 × 43 × 44 × 45, and by forming high order (up to 5^th order derivatives/6^th order tensors) Taylor series surrogates of the noise-whitened parameter-to-output map for a stochastic partial differential equation with boundary output.
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