On Recoverability of Randomly Compressed Tensors with Low CP Rank

Abstract: Our interest lies in the recoverability properties of compressed tensors under the \textit{canonical polyadic decomposition} (CPD) model. The considered problem is well-motivated in many applications, e.g., hyperspectral image and video compression. Prior work studied this problem under somewhat special assumptions---e.g., the latent factors of the tensor are sparse or drawn from absolutely continuous distributions. We offer an alternative result: We show that if the tensor is compressed by a subgaussian linear mapping, then the tensor is recoverable if the number of measurements is on the same order of magnitude as that of the model parameters---without strong assumptions on the latent factors. Our proof is based on deriving a \textit{restricted isometry property} (R.I.P.) under the CPD model via set covering techniques, and thus exhibits a flavor of classic compressive sensing. The new recoverability result enriches the understanding to the compressed CP tensor recovery problem; it offers theoretical guarantees for recovering tensors whose elements are not necessarily continuous or sparse.