On Recoverability of Randomly Compressed Tensors with Low CP Rank

Our interest lies in the recoverability properties of compressed tensors under the 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 a variety of assumptions, e.g., that the latent factors of the tensor are sparse and that the compressing matrix follows a joint absolutely continuous distribution. These results leverage analytical tools such as CPD uniqueness and algebraic geometry - which are elegant. In this work, 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. Unlike existing results, our proof is based on deriving a 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 sparse; the compressing matrix is also not restricted to continuous matrices under our framework. The newly derived covering number for tensors with low CP rank may also benefit future research, e.g., sketching based tensor compression for reducing computational burden.