Low-rank tensor completion: a Riemannian manifold preconditioning approach
This work addresses tensor completion, a problem in data analysis and machine learning for handling incomplete multi-dimensional data, with incremental improvements in optimization efficiency.
The authors tackled the tensor completion problem with rank constraint by proposing a Riemannian manifold preconditioning approach, resulting in algorithms that robustly outperform state-of-the-art methods across synthetic and real-world datasets.
We propose a novel Riemannian manifold preconditioning 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 proposed algorithms robustly outperform state-of-the-art algorithms across different synthetic and real-world datasets.