High-Order Tensor Recovery with A Tensor $U_1$ Norm

Jingjing Zheng, Wenzhe Wang, Xiaoqin Zhang, Yankai Cao, Xianta Jiang

Keywords: Tensor SVD; High Order Tensor Recovery; Tensor Completion

Abstract: Recently, numerous tensor SVD (t-SVD)-based tensor recovery methods have emerged, showing promise in processing visual data. However, these methods often suffer from performance degradation when confronted with high-order tensor data exhibiting non-smooth changes (possibly caused by random slice permutation), commonly observed in real-world scenarios but ignored by the traditional t-SVD-based methods. Our objective in this study is to provide an effective tensor recovery technique for handling non-smooth changes in tensor data and efficiently exploring the correlations of high-order tensor data across its various dimensions. To this end, we introduce a new tensor decomposition and a new tensor norm called the Tensor U1 norm. An optimization algorithm is proposed to solve the resulting tensor completion model iteratively by combining the proximal algorithm with the Alternating Direction Method of Multipliers. Theoretical analysis showed the convergence of the algorithm to the Karush–Kuhn–Tucker (KKT) point of the optimization problem. Numerical experiments demonstrated the effectiveness of the proposed method in high-order tensor completion, especially for tensor data with non-smooth changes. This study fills a critical gap in the t-SVD-based tensor recovery by providing a practical and effective solution that enables the exploration of correlations in high-order tensor data across its different dimensions, even in the presence of non-smooth changes.

Primary Area: general machine learning (i.e., none of the above)

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Submission Number: 7971