IEEE Trans Image Process. 2017 May;26(5):2466-2479. doi: 10.1109/TIP.2017.2672439. Epub 2017 Feb 20.
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 TT (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 TT (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.
本文提出了一种新的张量补全方法，用于恢复张量表示的数据中缺失的条目。该方法基于张量火车（TT）秩，由于其定义来自平衡的矩阵化方案，因此能够从张量中提取隐藏信息。相应地，提出了新的张量补全优化公式以及两种新的算法来求解这些公式。第一个算法称为通过 TT 的简单低秩张量补全（SiLRTC-TT），它与基于 TT 秩的核范数最小化密切相关。第二个算法是基于张量的多线性矩阵分解模型来近似张量的 TT 秩，称为通过 TT 的并行矩阵分解张量补全（TMac-TT）。还提出了一种张量扩充方案，将低阶张量转换为高阶张量，以增强 SiLRTC-TT 和 TMac-TT 的有效性。彩色图像和视频恢复的仿真结果表明，与所有其他方法相比，我们的方法具有明显的优势。
