New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition
This work addresses tensor completion problems for data analysis applications, presenting an incremental improvement in algorithmic efficiency.
The authors tackled tensor completion by proposing new Riemannian preconditioned algorithms using polyadic decomposition, which proved more memory- and time-efficient than state-of-the-art methods and showed greater tolerance for overestimated rank parameters.
We propose new Riemannian preconditioned algorithms for low-rank tensor completion via the polyadic decomposition of a tensor. These algorithms exploit a non-Euclidean metric on the product space of the factor matrices of the low-rank tensor in the polyadic decomposition form. This new metric is designed using an approximation of the diagonal blocks of the Hessian of the tensor completion cost function, thus has a preconditioning effect on these algorithms. We prove that the proposed Riemannian gradient descent algorithm globally converges to a stationary point of the tensor completion problem, with convergence rate estimates using the $Ł$ojasiewicz property. Numerical results on synthetic and real-world data suggest that the proposed algorithms are more efficient in memory and time compared to state-of-the-art algorithms. Moreover, the proposed algorithms display a greater tolerance for overestimated rank parameters in terms of the tensor recovery performance, thus enable a flexible choice of the rank parameter.