Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent

TL;DR

This paper introduces a preconditioned Riemannian gradient descent method for low-rank tensor completion in tensor train format, achieving faster convergence and reduced computational time in high-dimensional tensor recovery tasks.

Contribution

It proposes a novel PRGD algorithm with proven linear convergence for TT-format tensor completion, outperforming existing methods in speed and efficiency.

Findings

Reduced computation time by two orders of magnitude on simulated data

Significantly fewer iterations in hyperspectral image completion

Effective in quantum state tomography applications

Abstract

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.