Graph embedding and extensions: A general framework for dimensionality reduction
Graph embedding and extensions: A general framework for dimensionality reduction
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DOI：
10.1109/tpami.2007.250598
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发表时间：
2007-01-01
影响因子：
23.6
通讯作者：
Lin, Stephen
中科院分区：
文献类型：
--
作者：
Yan, Shuicheng;Xu, Dong;Lin, Stephen
在过去的几十年中，一大类算法——有监督的或无监督的；源于统计学或几何理论——被设计出来，为降维问题提供不同的解决方案。尽管这些算法的动机不同，但我们在本文中提出了一种称为图嵌入的通用公式，以便在一个通用框架内统一它们。在图嵌入中，每种算法都可被视为一个特定的内蕴图的直接图嵌入或其线性/核/张量扩展，该内蕴图描述了数据集的某些期望的统计或几何性质，并受到尺度归一化的约束，或者受到一个惩罚图的约束，该惩罚图表征了一种应该避免的统计或几何性质。此外，图嵌入框架可用作开发新的降维算法的通用平台。通过将此框架用作工具，我们提出了一种新的有监督降维算法，称为边际费舍尔分析，其中内蕴图表征类内紧凑性，并将每个数据点与其同类的相邻点相连，而惩罚图连接边际点并表征类间可分性。我们表明，由于数据分布假设和可用的投影方向，边际费舍尔分析有效地克服了传统线性判别分析算法的局限性。真实的人脸识别实验表明，与线性判别分析及其相应的核和张量扩展相比，我们提出的边际费舍尔分析具有优越性。
Over the past few decades, a large family of algorithms-supervised or unsupervised; stemming from statistics or geometry theory-has been designed to provide different solutions to the problem of dimensionality reduction. Despite the different motivations of these algorithms, we present in this paper a general formulation known as graph embedding to unify them within a common framework. In graph embedding, each algorithm can be considered as the direct graph embedding or its linear/kernel/tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set, with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided. Furthermore, the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms. By utilizing this framework as a tool, we propose a new supervised dimensionality reduction algorithm called Marginal Fisher Analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class, while the penalty graph connects the marginal points and characterizes the interclass separability. We show that MFA effectively overcomes the limitations of the traditional Linear Discriminant Analysis algorithm due to data distribution assumptions and available projection directions. Real face recognition experiments show the superiority of our proposed MFA in comparison to LDA, also for corresponding kernel and tensor extensions.