This paper proposes a novel approach to tensor completion, which recovers
missing entries of data represented by tensors. The approach is based on the
tensor train (TT) rank, which is able to capture hidden information from
tensors thanks to its definition from a well-balanced matricization scheme.
Accordingly, new optimization formulations for tensor completion are proposed
as well as two new algorithms for their solution. The first one called simple
low-rank tensor completion via tensor train (SiLRTC-TT) is intimately related
to minimizing a nuclear norm based on TT rank. The second one is from a
multilinear matrix factorization model to approximate the TT rank of a tensor,
and is called tensor completion by parallel matrix factorization via tensor
train (TMac-TT). A tensor augmentation scheme of transforming a low-order
tensor to higher-orders is also proposed to enhance the effectiveness of
SiLRTC-TT and TMac-TT. Simulation results for color image and video recovery
show the clear advantage of our method over all other methods.

本文提出一种新颖的张量完成方法，基于张量列车 (TT) 秩，提出了新的优化公式和算法。其中，SiLRTC-TT 是通过在 TT 秩的基础上最小化核范数实现的，TMac-TT 是通过基于多线性矩阵分解来逼近张量 TT 秩。文章还提出了一种向高阶张量转换的方法以增强 SiLRTC-TT 和 TMac-TT 的有效性。仿真结果表明该方法在彩色图像和视频恢复方面具有明显优势。