  This paper proposes a novel formulation of the tensor completion problem to
impute missing entries of data represented by tensors. The formulation is
introduced in terms of tensor train (TT) rank which can effectively capture
global information of tensors thanks to its construction by a well-balanced
matricization scheme. Two algorithms are proposed to solve the corresponding
tensor completion problem. The first one called simple low-rank tensor
completion via tensor train (SiLRTC-TT) is intimately related to minimizing the
TT nuclear norm. The second one is based on a multilinear matrix factorization
model to approximate the TT rank of the tensor and called tensor completion by
parallel matrix factorization via tensor train (TMac-TT). These algorithms are
applied to complete both synthetic and real world data tensors. Simulation
results of synthetic data show that the proposed algorithms are efficient in
estimating missing entries for tensors with either low Tucker rank or TT rank
while Tucker-based algorithms are only comparable in the case of low Tucker
rank tensors. When applied to recover color images represented by ninth-order
tensors augmented from third-order ones, the proposed algorithms outperforms
the Tucker-based algorithms.
