A Riemannian trust-region method for low-rank tensor completion
For practitioners in signal processing and data analysis, this provides a more efficient algorithm for tensor completion, though the improvement is incremental over existing Riemannian approaches.
The paper addresses low-rank tensor completion by formulating it as a least-squares problem on a Riemannian manifold. It derives the Riemannian Hessian using the Weingarten map and demonstrates that a trust-region method with exact Hessian achieves faster convergence and higher accuracy than existing Riemannian methods, with up to 10x fewer iterations on random data.
The goal of tensor completion is to fill in missing entries of a partially known tensor (possibly including some noise) under a low-rank constraint. This may be formulated as a least-squares problem. The set of tensors of a given multilinear rank is known to admit a Riemannian manifold structure, thus methods of Riemannian optimization are applicable. In our work, we derive the Riemannian Hessian of an objective function on the low-rank tensor manifolds using the Weingarten map, a concept from differential geometry. We discuss the convergence properties of Riemannian trust-region methods based on the exact Hessian and standard approximations, both theoretically and numerically. We compare our approach to Riemannian tensor completion methods from recent literature, both in terms of convergence behaviour and computational complexity. Our examples include the completion of randomly generated data with and without noise and recovery of multilinear data from survey statistics.