Trace Ratio vs. Ratio Trace for Dimensionality Reduction
Trace Ratio vs. Ratio Trace for Dimensionality Reduction
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DOI：
10.1109/cvpr.2007.382983
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发表时间：
2007-06
期刊：
影响因子：
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通讯作者：
Huan Wang;Shuicheng Yan;Dong Xu;Xiaoou Tang;Thomas S. Huang
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文献类型：
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作者：
Huan Wang;Shuicheng Yan;Dong Xu;Xiaoou Tang;Thomas S. Huang
用于降维的一大类算法以求解arg maxW Tr（WT SPW）/Tr（WT SIW）1形式的迹比问题结束，其通常被变换成对应的比率迹形式arg maxW Tr[（WTSIW）-1（WTSPW）]以获得封闭形式但不精确的解。在这项工作中，一个有效的迭代过程中，直接解决跟踪比问题。在每个步骤中，用从前一步骤计算的迹比率值λ来求解迹差问题arg maxW Tr [WT（SP -λ daSI）W]。投影矩阵W的收敛性，以及跟踪比值λ的全局最优性，证明了基于点到集映射理论。此外，该程序进一步扩展到解决更一般的约束WTCW=I的迹比问题，并提供基于核的子空间学习问题的精确解。人脸和UCI数据上的大量实验表明，所提出的解决方案的高收敛速度，以及其在分类能力的优势，比相应的解决方案的比例跟踪问题。
A large family of algorithms for dimensionality reduction end with solving a Trace Ratio problem in the form of arg maxW Tr(WT SPW)/Tr(WT SIW)1, which is generally transformed into the corresponding Ratio Trace form arg maxW Tr[ (WTSIW)-1 (WTSPW) ] for obtaining a closed-form but inexact solution. In this work, an efficient iterative procedure is presented to directly solve the Trace Ratio problem. In each step, a Trace Difference problem arg maxW Tr [WT (SP - lambdaSI) W] is solved with lambda being the trace ratio value computed from the previous step. Convergence of the projection matrix W, as well as the global optimum of the trace ratio value lambda, are proven based on point-to-set map theories. In addition, this procedure is further extended for solving trace ratio problems with more general constraint WTCW=I and providing exact solutions for kernel-based subspace learning problems. Extensive experiments on faces and UCI data demonstrate the high convergence speed of the proposed solution, as well as its superiority in classification capability over corresponding solutions to the ratio trace problem.