Horizontal Dimensionality Reduction and Iterated Frame Bundle Development
Horizontal Dimensionality Reduction and Iterated Frame Bundle Development
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水平降维和迭代框架束开发
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
S. Sommer
中科院分区：
文献类型：
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作者：
S. Sommer
在欧氏向量空间中，降维可以以数据均值为中心。相比之下，距离不会拆分为正交分量，并且居中分析会在存在曲率的情况下扭曲点间距离。在这篇文章中，我们定义了黎曼流形中数据的降维过程，它将分析从中心点移动到局部距离测量。水平分量分析测量相对于低阶水平分量的距离，提供多峰分布和随机过程产生的数据的自然视图。我们通过迭代水平展开将非局部低维子空间参数化，这是一种将测地线子空间和多项式子空间推广到黎曼流形的构造性过程。文中给出了低维水平分量如何成功逼近多峰分布的例子。
In Euclidean vector spaces, dimensionality reduction can be centered at the data mean. In contrast, distances do not split into orthogonal components and centered analysis distorts inter-point distances in the presence of curvature. In this paper, we define a dimensionality reduction procedure for data in Riemannian manifolds that moves the analysis from a center point to local distance measurements. Horizontal component analysis measures distances relative to lower-order horizontal components providing a natural view of data generated by multimodal distributions and stochastic processes. We parametrize the non-local, low-dimensional subspaces by iterated horizontal development, a constructive procedure that generalizes both geodesic subspaces and polynomial subspaces to Riemannian manifolds. The paper gives examples of how low-dimensional horizontal components successfully approximate multimodal distributions.