摘要翻译:
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:从不同的起点出发,迭代改进的解会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。本文给出了对这一问题进行全面研究的结果。我们发现,虽然所有的张量分解算法在随机数据和严重置乱数据上都不能达到唯一的全局解;然而,令人惊讶的是,在现实生活中的多个数据集(即使存在大量的置乱和遮挡)上,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则产生非唯一的解。我们给出了一个基于特征值的判定解唯一性的准则。
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英文标题:
《Are Tensor Decomposition Solutions Unique? On the global convergence of
HOSVD and ParaFac algorithms》
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作者:
Dijun Luo, Heng Huang, Chris Ding
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最新提交年份:
2009
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分类信息:
一级分类:Computer Science 计算机科学
二级分类:Computer Vision and Pattern Recognition 计算机视觉与模式识别
分类描述:Covers image processing, computer vision, pattern recognition, and scene understanding. Roughly includes material in ACM Subject Classes I.2.10, I.4, and I.5.
涵盖图像处理、计算机视觉、模式识别和场景理解。大致包括ACM课程I.2.10、I.4和I.5中的材料。
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一级分类:Computer Science 计算机科学
二级分类:Artificial Intelligence 人工智能
分类描述:Covers all areas of AI except Vision, Robotics, Machine Learning, Multiagent Systems, and Computation and Language (Natural Language Processing), which have separate subject areas. In particular, includes Expert Systems, Theorem Proving (although this may overlap with Logic in Computer Science), Knowledge Representation, Planning, and Uncertainty in AI. Roughly includes material in ACM Subject Classes I.2.0, I.2.1, I.2.3, I.2.4, I.2.8, and I.2.11.
涵盖了人工智能的所有领域,除了视觉、机器人、机器学习、多智能体系统以及计算和语言(自然语言处理),这些领域有独立的学科领域。特别地,包括专家系统,定理证明(尽管这可能与计算机科学中的逻辑重叠),知识表示,规划,和人工智能中的不确定性。大致包括ACM学科类I.2.0、I.2.1、I.2.3、I.2.4、I.2.8和I.2.11中的材料。
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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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PDF链接:
https://arxiv.org/pdf/0902.4521