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.

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.