Riemannian Manifold Learning

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 lowdimensional space. Experiments on synthetic data, as well as real-world im...

• Type: Journal
• Year: 2008
• Where: PAMI
• Authors: Tong Lin, Hongbin Zha
• Keywords: Low-dimensional Riemannian Manifold, Manifold, PAMI 2008, Riemannian Geometry