Tensor Completion via Tensor Train Based Low‐Rank Quotient Geometry Under a Preconditioned Metric

ABSTRACT The low‐rank tensor completion problem is about recovering a tensor from partially observed entries. We consider this problem in the tensor train format and extend the preconditioned metric from the matrix case to the tensor case. The first‐order and second‐order quotient geometry of the manifold of fixed tensor train rank tensors under this metric is studied in detail. Algorithms, including Riemannian gradient descent, Riemannian conjugate gradient, and Riemannian Gauss–Newton, have been proposed for the tensor completion problem based on the quotient geometry. It has also been shown that the Riemannian Gauss–Newton method on the quotient geometry is equivalent to the Riemannian Gauss–Newton method on the embedded geometry with a specific retraction. Empirical evaluations on random instances as well as on function‐related tensors show that the proposed algorithms are competitive with other existing algorithms in terms of recovery ability, convergence performance, and reconstruction quality.