Tensor Completion using Subspace Information

Under standard uniform sampling, existing polynomial-time guarantees generally require more observations than the number of degree of freedom, motivating the study of a possible statistical-to-computational gap in highly missing regimes. Fortunately, in many practical scenarios, side information is available, which can provide valuable insights to mitigate these challenges.

In this paper, we introduce an algorithm called Tensor Completion using Subspace Information (TCSI) that incorporates side information through an estimated subspace. Our approach first extracts the subspace from the available side information and then reformulates tensor completion as a matrix regression problem. We provide a theoretical analysis showing that, when accurate subspace information is available, the required sample complexity is reduced to nearly linear order in the uncoupled ambient dimensions, removing the coupled-mode dimension from the leading term.