Fast similarity computation in factorized tensors

Low-rank factorizations of higher-order tensors have become an invaluable tool for researchers from many scientific disciplines. Tensor factorizations have been successfully applied for moderately sized multimodal data sets involving a small number of modes. However, a significant hindrance to the full realization of the potential of tensor methods is a lack of scalability on the client side: even when low-rank representations are provided by an external agent possessing the necessary computational resources, client applications are quickly rendered infeasible by the space requirements for explicitly storing a (dense) low-rank representation of the input tensor. We consider the problem of efficiently computing common similarity measures between entities expressed by fibers (vectors) or slices (matrices) within a given factorized tensor. We show that after appropriate preprocessing, inner products can be efficiently computed independently of the dimensions of the input tensor.