Riemannian conjugate gradient descent method for fixed multi rank third-order tensor completion

The goal of tensor completion is to fill in missing entries of a partially known tensor under a low-rank constraint. In this paper, we study low rank third-order tensor completion problems by using Riemannian optimization methods on the smooth manifold. Here the tensor rank is defined to be a multi-rank under the Discrete Cosine Transform-related transform tensor-tensor product. With suitable incoherence conditions on the underlying low multi-rank rank tensor, we show that the proposed Riemannian optimization method is guaranteed to converge to the underlying low multi-rank tensor with a high probability. Number of sampling entries required for convergence are also derived. Numerical examples of synthetic data and real data sets are reported to demonstrate that the performance of the proposed method is better than that of tensor-based method using the Tucker-rank model in terms of computational time, and that of the matrix-based completion method in terms of number of sampling entries.