Riemannian manifold learning
Riemannian manifold learning
复制标题
黎曼流形学习
DOI：
10.1109/tpami.2007.70735
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发表时间：
2008-05-01
影响因子：
23.6
通讯作者：
Zha, Hongbin
中科院分区：
文献类型：
--
作者：
Lin, Tong;Zha, Hongbin
最近，流形学习在模式识别、数据分析和机器学习中得到了广泛应用。本文提出了一种新的框架，称为黎曼流形学习（RML），它基于这样一种假设：输入的高维数据位于一个本质上低维的黎曼流形上。其主要思想是将降维问题表述为黎曼几何中的一个经典问题，即如何为给定的黎曼流形构建坐标图？我们针对一组无组织的数据点实现了黎曼法坐标图，它在黎曼几何中应用最为广泛。首先，基于对底层流形的一种高效单纯形重构，估计两个输入参数（邻域大小\(k\)和本征维数\(d\)）。然后，计算法坐标以将输入的高维数据映射到低维空间。在合成数据以及真实世界图像上的实验表明，我们的算法能够学习数据的内在几何结构，保持径向测地距离，并产生规则的嵌入。
Recently, manifold learning has been widely exploited in pattern recognition, data analysis, and machine learning. This paper presents a novel framework, called Riemannian manifold learning (RML), based on the assumption that the input high-dimensional data lie on an intrinsically low-dimensional Riemannian manifold. The main idea is to formulate the dimensionality reduction problem as a classical problem in Riemannian geometry, that is, how to construct coordinate charts for a given Riemannian manifold? We implement the Riemannian normal coordinate chart, which has been the most widely used in Riemannian geometry, for a set of unorganized data points. First, two input parameters (the neighborhood size k and the intrinsic dimension d) are estimated based on an efficient simplicial reconstruction of the underlying manifold. Then, the normal coordinates are computed to map the input high-dimensional data into a low- dimensional space. Experiments on synthetic data, as well as real-world images, demonstrate that our algorithm can learn intrinsic geometric structures of the data, preserve radial geodesic distances, and yield regular embeddings.