Cost-efficient Gaussian tensor network embeddings for tensor-structured inputs

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.