Generalized Low-Rank Approximations of Matrices Revisited
Generalized Low-Rank Approximations of Matrices Revisited
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重新审视矩阵的广义低阶近似
DOI：
10.1109/tnn.2010.2040290
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发表时间：
2010-04-01
影响因子：
--
通讯作者：
Tan, Xiaoyang
中科院分区：
文献类型：
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作者：
Liu, Jun;Chen, Songcan;Tan, Xiaoyang
与奇异值分解（SVD）相比，广义低秩矩阵近似（GLRAM）可以消耗更少的计算时间，获得更高的压缩比，并产生有竞争力的分类性能。 GLRAM已成功应用于图像压缩和检索等应用，并相继提出了相当多的扩展。然而，在文献中，GLRAM 的一些基本特性和关键问题尚未被探索或解决。为此，我们在本文中重新审视 GLRAM。首先，我们揭示了 GLRAM 和 SVD 之间如此密切的关系，除了施加的约束之外，GLRAM 的目标函数与 SVD 的目标函数相同。其次，我们推导出 GLRAM 目标函数的下界，并讨论何时可以触及下界。此外，从最小化下界的角度来看，我们回答了叶提出的一个开放问题（机器学习，2005），即对实验现象的理论证明，即在给定的降维数下，当左右变换具有相同的列数时，获得最低的重建误差。第三，我们探讨GLRAM何时以及为何能够在压缩方面表现良好，这是关系到GLRAM可用性的一个基本问题。
Compared to singular value decomposition (SVD), generalized low-rank approximations of matrices (GLRAM) can consume less computation time, obtain higher compression ratio, and yield competitive classification performance. GLRAM has been successfully applied to applications such as image compression and retrieval, and quite a few extensions have been successively proposed. However, in literature, some basic properties and crucial problems with regard to GLRAM have not been explored or solved yet. For this sake, we revisit GLRAM in this paper. First, we reveal such a close relationship between GLRAM and SVD that GLRAM's objective function is identical to SVD's objective function except the imposed constraints. Second, we derive a lower bound of GLRAM's objective function, and discuss when the lower bound can be touched. Moreover, from the viewpoint of minimizing the lower bound, we answer one open problem raised by Ye (Machine Learning, 2005), i.e., a theoretical justification of the experimental phenomenon that, under given number of reduced dimension, the lowest reconstruction error is obtained when the left and right transformations have equal number of columns. Third, we explore when and why GLRAM can perform well in terms of compression, which is a fundamental problem concerning the usability of GLRAM.