在线线性、鲁棒和非线性子空间学习的内禀Grassmann算法
Intrinsic Grassmann Averages for Online Linear, Robust and Nonlinear Subspace Learning
摘要：
主成分分析（PCA）和核主成分分析（KPCA）是机器学习中降维的基本方法。前者是在有限维中找到这种近似的技术，后者通常是在无限维再生核希尔伯特空间（RKHS）中。在本文中，我们提出了一个几何框架计算的主要线性子空间在这两种情况下，以及强大的PCA的情况下，这相当于计算的内在平均空间的所有子空间：格拉斯曼流形。这个流形上的点被定义为观测的$K$ -元组所张成的子空间。这些子空间的内在格拉斯曼平均值与观测的主成分一致，当它们是从高斯分布中得出的。我们在RKHS的情况下，类似的结果，并提供了一个有效的算法计算到这个平均子空间的投影。其结果是一种类似于KPCA的方法，其速度要快得多。此外，我们提出了一个新的在线版本的KPCA使用我们的几何框架。我们所有的算法的竞争力表现证明了各种真实和合成数据集。
英文摘要：
Principal Component Analysis (PCA) and Kernel Principal Component Analysis (KPCA) are fundamental methods in machine learning for dimensionality reduction. The former is a technique for finding this approximation in finite dimensions and the latter is often in an infinite dimensional Reproducing Kernel Hilbert-space (RKHS). In this paper, we present a geometric framework for computing the principal linear subspaces in both situations as well as for the robust PCA case, that amounts to computing the intrinsic average on the space of all subspaces: the Grassmann manifold. Points on this manifold are defined as the subspaces spanned by $K$-tuples of observations. The intrinsic Grassmann average of these subspaces are shown to coincide with the principal components of the observations when they are drawn from a Gaussian distribution. We show similar results in the RKHS case and provide an efficient algorithm for computing the projection onto the this average subspace. The result is a method akin to KPCA which is substantially faster. Further, we present a novel online version of the KPCA using our geometric framework. Competitive performance of all our algorithms are demonstrated on a variety of real and synthetic data sets.
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