We propose Riemannian preconditioned algorithms for the tensor completion problem via tensor ring decomposition. A new Riemannian metric is developed on the product space of the mode-2 unfolding matrices of the core tensors in tensor ring decomposition. The construction of this metric aims to approximate the Hessian of the cost function by its diagonal blocks, paving the way for various Riemannian optimization methods. Specifically, we propose the Riemannian gradient descent and Riemannian conjugate gradient algorithms. We prove that both algorithms globally converge to a stationary point. In the implementation, we exploit the tensor structure and adopt an economical procedure to avoid large matrix formulation and computation in gradients, which significantly reduces the computational cost. Numerical experiments on various synthetic and real-world datasets -- movie ratings, hyperspectral images, and high-dimensional functions -- suggest that the proposed algorithms are more efficient and have better reconstruction ability than other candidates.

翻译：我们针对张量环分解的张量补全问题提出了黎曼预条件算法。在张量环分解的核心张量的模式-2展开矩阵乘积空间上，我们发展了一种新的黎曼度量。该度量的构造旨在通过代价函数的对角块近似其黑塞矩阵，为多种黎曼优化方法奠定基础。具体而言，我们提出了黎曼梯度下降法和黎曼共轭梯度法，并证明这两种算法均可全局收敛至稳定点。在实现过程中，我们利用张量结构并采用经济型流程，避免梯度计算中的大规模矩阵构造与运算，从而显著降低计算成本。在多种合成数据集与真实世界数据集（包括电影评分、高光谱图像及高维函数）上的数值实验表明，所提算法相比其他候选方法具有更高的效率与更优的重构能力。