2022/09/11 by Jian-Feng Cai, Cai, Jian-Feng, Wen Huang +5
Engineering · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #FOS: Mathematics #Medical Imaging and Analysis #Optimization and Control (math.OC) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2209.04786
openalex publication_date 2022/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper investigates the low-rank tensor completion problem, which is about recovering a tensor from partially observed entries. We consider this problem in the tensor train format and extend the preconditioned metric from the matrix case to the tensor case. The first-order and second-order quotient geometry of the manifold of fixed tensor train rank tensors under this metric is studied in detail. Algorithms, including Riemannian gradient descent, Riemannian conjugate gradient, and Riemannian Gauss-Newton, have been proposed for the tensor completion problem based on the quotient geometry. It has also been shown that the Riemannian Gauss-Newton method on the quotient geometry is equivalent to the Riemannian Gauss-Newton method on the embedded geometry with a specific retraction. Empirical evaluations on random instances as well as on function-related tensors show that the proposed algorithms are competitive with other existing algorithms in terms of recovery ability, convergence performance, and reconstruction quality.