Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent
This work addresses tensor completion for fields like quantum computing and image processing, offering a faster method that is incremental in improving computational efficiency.
The paper tackled low-rank tensor completion using a preconditioned Riemannian gradient descent algorithm (PRGD) for the tensor train format, achieving linear convergence and reducing computation time by two orders of magnitude on simulated datasets while significantly cutting iterations in real applications like hyperspectral image completion.
Low-rank tensor completion aims to recover a tensor from partially observed entries, and it is widely applicable in fields such as quantum computing and image processing. Due to the significant advantages of the tensor train (TT) format in handling structured high-order tensors, this paper investigates the low-rank tensor completion problem based on the TT-format. We proposed a preconditioned Riemannian gradient descent algorithm (PRGD) to solve low TT-rank tensor completion and establish its linear convergence. Experimental results on both simulated and real datasets demonstrate the effectiveness of the PRGD algorithm. On the simulated dataset, the PRGD algorithm reduced the computation time by two orders of magnitude compared to existing classical algorithms. In practical applications such as hyperspectral image completion and quantum state tomography, the PRGD algorithm significantly reduced the number of iterations, thereby substantially reducing the computational time.