**Keywords:**tensor networks, randomized numerical linear algebra, non-convex optimization, tensor decomposition, tensor train

**TL;DR:**We provide a systematic way to design tensor network embedding consisting of Gaussian random tensors, such that multiplying the embedding with the input has low computational cost.

**Abstract:**This work discusses tensor network embeddings, which are random matrices ($S$) with tensor network structure. These embeddings have been used to perform dimensionality reduction of tensor network structured inputs $x$ and accelerate applications such as tensor decomposition and kernel regression. Existing works have designed embeddings for inputs $x$ with specific structures, such as the Kronecker product or Khatri-Rao product, such that the computational cost for calculating $Sx$ is efficient. We provide a systematic way to design tensor network embeddings consisting of Gaussian random tensors, such that for inputs with more general tensor network structures, both the sketch size (row size of $S$) and the sketching computational cost are low. We analyze general tensor network embeddings that can be reduced to a sequence of sketching matrices. We provide a sufficient condition to quantify the accuracy of such embeddings and derive sketching asymptotic cost lower bounds using embeddings that satisfy this condition and have a sketch size lower than any input dimension. We then provide an algorithm to efficiently sketch input data using such embeddings. The sketch size of the embedding used in the algorithm has a linear dependence on the number of sketching dimensions of the input. Assuming tensor contractions are performed with classical dense matrix multiplication algorithms, this algorithm achieves asymptotic cost within a factor of $O(\sqrt{m})$ of our cost lower bound, where $m$ is the sketch size. Further, when each tensor in the input has a dimension that needs to be sketched, this algorithm yields the optimal sketching asymptotic cost. We apply our sketching analysis to inexact tensor decomposition optimization algorithms. We provide a sketching algorithm for CP decomposition that is asymptotically faster than existing work in multiple regimes, and show the optimality of an existing algorithm for tensor train rounding.

**Supplementary Material:**pdf

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