  Tensor completion is a technique of filling missing elements of the
incomplete data tensors. It being actively studied based on the convex
optimization scheme such as nuclear-norm minimization. When given data tensors
include some noises, the nuclear-norm minimization problem is usually converted
to the nuclear-norm `regularization' problem which simultaneously minimize
penalty and error terms with some trade-off parameter. However, the good value
of trade-off is not easily determined because of the difference of two units
and the data dependence. In the sense of trade-off tuning, the noisy tensor
completion problem with the `noise inequality constraint' is better choice than
the `regularization' because the good noise threshold can be easily bounded
with noise standard deviation. In this study, we tackle to solve the convex
tensor completion problems with two types of noise inequality constraints:
Gaussian and Laplace distributions. The contributions of this study are
follows: (1) New tensor completion and denoising models using tensor total
variation and nuclear-norm are proposed which can be characterized as a
generalization/extension of many past matrix and tensor completion models, (2)
proximal mappings for noise inequalities are derived which are analytically
computable with low computational complexity, (3) convex optimization algorithm
is proposed based on primal-dual splitting framework, (4) new step-size
adaptation method is proposed to accelerate the optimization, and (5) extensive
experiments demonstrated the advantages of the proposed method for visual data
retrieval such as for color images, movies, and 3D-volumetric data.
