  In this paper, we propose a novel tensor learning and coding model for
third-order data completion. Our model is to learn a data-adaptive dictionary
from the given observations, and determine the coding coefficients of
third-order tensor tubes. In the completion process, we minimize the
low-rankness of each tensor slice containing the coding coefficients. By
comparison with the traditional pre-defined transform basis, the advantages of
the proposed model are that (i) the dictionary can be learned based on the
given data observations so that the basis can be more adaptively and accurately
constructed, and (ii) the low-rankness of the coding coefficients can allow the
linear combination of dictionary features more effectively. Also we develop a
multi-block proximal alternating minimization algorithm for solving such tensor
learning and coding model, and show that the sequence generated by the
algorithm can globally converge to a critical point. Extensive experimental
results for real data sets such as videos, hyperspectral images, and traffic
data are reported to demonstrate these advantages and show the performance of
the proposed tensor learning and coding method is significantly better than the
other tensor completion methods in terms of several evaluation metrics.
