# Precon tensor completion

### Low-rank tensor completion: a Riemannian manifold preconditioning approach

#### Authors

H. Kasai and B. Mishra

#### Abstract

We propose a novel Riemannian manifold pre- conditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry that exists in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop preconditioned nonlinear conjugate gradient and stochastic gradient descent algorithms for batch and online setups, respectively. Concrete matrix representations of various optimization-related ingredients are listed. Numerical comparisons suggest that our pro- posed algorithms robustly outperform state-of-the-art algorithms across different synthetic and real-world datasets.

 Small scale: rank is (10,10,10). Large scale: over sampling (OS) ratio is 10. Noise: size is (10000, 10000, 10000), rank is (5,5,5), OS is 10, and ${\Gamma}$ is the test set. Ribeira hypersectral image: observation ratio is 5% and ${\Gamma}$ is the test set.