  In recent studies, the tensor ring (TR) rank has shown high effectiveness in
tensor completion due to its ability of capturing the intrinsic structure
within high-order tensors. A recently proposed TR rank minimization method is
based on the convex relaxation by penalizing the weighted sum of nuclear norm
of TR unfolding matrices. However, this method treats each singular value
equally and neglects their physical meanings, which usually leads to suboptimal
solutions in practice. In this paper, we propose to use the logdet-based
function as a nonconvex smooth relaxation of the TR rank for tensor completion,
which can more accurately approximate the TR rank and better promote the
low-rankness of the solution. To solve the proposed nonconvex model
efficiently, we develop an alternating direction method of multipliers
algorithm and theoretically prove that, under some mild assumptions, our
algorithm converges to a stationary point. Extensive experiments on color
images, multispectral images, and color videos demonstrate that the proposed
method outperforms several state-of-the-art competitors in both visual and
quantitative comparison. Key words: nonconvex optimization, tensor ring rank,
logdet function, tensor completion, alternating direction method of
multipliers.
