  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.
