张量分解解是唯一的吗？关于HOSVD和ParaFac算法的全局收敛性
Are Tensor Decomposition Solutions Unique? On the global convergence of HOSVD and ParaFac algorithms
摘要：
对于HOSVD和ParaFac等张量分解，目标函数是非凸的。这意味着，理论上，存在大量的局部最优解：从不同的起点开始，迭代改进的解将收敛到不同的局部解。这种非唯一性给图像压缩和检索带来了稳定性和可靠性问题。在本文中，我们提出了一个全面的调查这个问题的结果。我们发现，尽管所有张量分解算法在随机数据和严重扰乱的数据上都不能得到唯一的全局解;然而令人惊讶的是，在所有现实生活中的几个数据集上（即使有大量的扰乱和遮挡），HOSVD总是在适合实际应用的参数区域内产生唯一的全局解，而ParaFac产生非唯一的解。我们提供了一个基于特征值的规则来评估解决方案的唯一性。
英文摘要：
For tensor decompositions such as HOSVD and ParaFac, the objective functions are nonconvex. This implies, theoretically, there exists a large number of local optimas: starting from different starting point, the iteratively improved solution will converge to different local solutions. This non-uniqueness present a stability and reliability problem for image compression and retrieval. In this paper, we present the results of a comprehensive investigation of this problem. We found that although all tensor decomposition algorithms fail to reach a unique global solution on random data and severely scrambled data; surprisingly however, on all real life several data sets (even with substantial scramble and occlusions), HOSVD always produce the unique global solution in the parameter region suitable to practical applications, while ParaFac produce non-unique solutions. We provide an eigenvalue based rule for the assessing the solution uniqueness.
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