  We aim to provably complete a sparse and highly-missing tensor in the
presence of covariate information along tensor modes. Our motivation comes from
online advertising where users click-through-rates (CTR) on ads over various
devices form a CTR tensor that has about 96% missing entries and has many zeros
on non-missing entries, which makes the standalone tensor completion method
unsatisfactory. Beside the CTR tensor, additional ad features or user
characteristics are often available. In this paper, we propose
Covariate-assisted Sparse Tensor Completion (COSTCO) to incorporate covariate
information for the recovery of the sparse tensor. The key idea is to jointly
extract latent components from both the tensor and the covariate matrix to
learn a synthetic representation. Theoretically, we derive the error bound for
the recovered tensor components and explicitly quantify the improvements on
both the reveal probability condition and the tensor recovery accuracy due to
covariates. Finally, we apply COSTCO to an advertisement dataset consisting of
a CTR tensor and ad covariate matrix, leading to 23% accuracy improvement over
the baseline. An important by-product is that ad latent components from COSTCO
reveal interesting ad clusters, which are useful for better ad targeting.
