Riemannian preconditioning for tensor completion
This work addresses tensor completion, a problem in data analysis and machine learning, by introducing a novel optimization method, though it appears incremental as it builds on existing Riemannian optimization frameworks.
The authors tackled the tensor completion problem with rank constraint by proposing a Riemannian preconditioning approach, resulting in a new algorithm that robustly outperforms state-of-the-art methods across synthetic and real-world datasets.
We propose a novel Riemannian preconditioning approach for the tensor completion problem with rank constraint. A Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop a preconditioned nonlinear conjugate gradient algorithm for the problem. To this end, concrete matrix representations of various optimization-related ingredients are listed. Numerical comparisons suggest that our proposed algorithm robustly outperforms state-of-the-art algorithms across different problem instances encompassing various synthetic and real-world datasets.