改进张量补全中的秩最小化方法，提升恢复精度。
Revisiting Trace Norm Minimization for Tensor Tucker Completion: A Direct Multilinear Rank Learning Approach
- 直接对等价表示的因子矩阵做迹范数最小化
- 实验显示多线性秩学习效果显著提升
- 适合需要高精度张量补全的研究者
为高效用Tucker格式表达张量数据，关键任务是降低多线性秩以避免模型过拟合。由于张量缺乏秩最小化工具，现有工作将Tucker多线性秩最小化转化为从张量展开的矩阵的迹范数最小化。然而本文揭示，现有基于迹范数的Tucker补全方法在多线性秩最小化上效率低下。为此，提出一种新的Tucker格式解释：将迹范数最小化应用于等价表示的因子矩阵，而非张量展开后的矩阵。基于新公式，提出固定点迭代算法并证明其收敛性。数值结果表明，所提算法在多线性秩学习和张量信号恢复精度方面显著优于现有迹范数方法。
原文摘要 · Abstract (English)
To efficiently express tensor data using the Tucker format, a critical task is to minimize the multilinear rank such that the model would not be over-flexible and lead to overfitting. Due to the lack of rank minimization tools in tensor, existing works connect Tucker multilinear rank minimization to trace norm minimization of matrices unfolded from the tensor data. While these formulations try to exploit the common aim of identifying the low-dimensional structure of the tensor and matrix, this paper reveals that existing trace norm-based formulations in Tucker completion are inefficient in multilinear rank minimization. We further propose a new interpretation of Tucker format such that trace norm minimization is applied to the factor matrices of the equivalent representation, rather than some matrices unfolded from tensor data. Based on the newly established problem formulation, a fixed point iteration algorithm is proposed, and its convergence is proved. Numerical results are presented to show that the proposed algorithm exhibits significant improved performance in terms of multilinear rank learning and consequently tensor signal recovery accuracy, compared to existing trace norm based Tucker completion methods.
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