基于格拉斯曼扩散映射的高维数据降维与分类
Grassmannian diffusion maps based dimension reduction and classification for high-dimensional data
摘要：
这项工作介绍了格拉斯曼扩散映射，一种新的非线性降维技术，通过将点表示为对应于格拉斯曼流形上的点的低维子空间来定义点之间的亲和力。该方法是专为应用程序，如图像识别和基于数据的分类，高维数据，可以在一个较低的维子空间compensately表示。GDMaps由两个阶段组成。首先是逐点线性降维，其中每个高维对象被映射到格拉斯曼。第二阶段是一个多点的非线性基于核的降维，使用扩散映射来识别格拉斯曼流形上的点的子空间结构。为了达到这个目的，一个适当的格拉斯曼核被用来构造一个图上的随机游动的转移矩阵，该图连接格拉斯曼流形上的点。转移矩阵的谱分析产生低维格拉斯曼扩散坐标，将数据嵌入低维再生核希尔伯特空间中。此外，一种新的数据分类/识别技术的基础上，建设的过完备字典的减少的维度，其原子是由格拉斯曼扩散坐标。考虑三个例子。首先，一个“玩具”的例子表明，GDMaps可以识别一个适当的参数化的单位球面上的结构点。第二个例子展示了GDMaps揭示高维随机场数据的内在子空间结构的能力。在最后一个例子中，解决了人脸识别问题，考虑到人脸图像受到不同的照明条件，在面部表情的变化，并发生闭塞。
英文摘要：
This work introduces the Grassmannian Diffusion Maps, a novel nonlinear dimensionality reduction technique that defines the affinity between points through their representation as low-dimensional subspaces corresponding to points on the Grassmann manifold. The method is designed for applications, such as image recognition and data-based classification of high-dimensional data that can be compactly represented in a lower dimensional subspace. The GDMaps is composed of two stages. The first is a pointwise linear dimensionality reduction wherein each high-dimensional object is mapped onto the Grassmann. The second stage is a multi-point nonlinear kernel-based dimension reduction using Diffusion maps to identify the subspace structure of the points on the Grassmann manifold. To this aim, an appropriate Grassmannian kernel is used to construct the transition matrix of a random walk on a graph connecting points on the Grassmann manifold. Spectral analysis of the transition matrix yields low-dimensional Grassmannian diffusion coordinates embedding the data into a low-dimensional reproducing kernel Hilbert space. Further, a novel data classification/recognition technique is developed based on the construction of an overcomplete dictionary of reduced dimension whose atoms are given by the Grassmannian diffusion coordinates. Three examples are considered. First, a "toy" example shows that the GDMaps can identify an appropriate parametrization of structured points on the unit sphere. The second example demonstrates the ability of the GDMaps to reveal the intrinsic subspace structure of high-dimensional random field data. In the last example, a face recognition problem is solved considering face images subject to varying illumination conditions, changes in face expressions, and occurrence of occlusions.
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