CSDL Home IEEE Transactions on Pattern Analysis & Machine Intelligence 2008 vol.30 Issue No.05  May
Subscribe
Issue No.05  May (2008 vol.30)
pp: 796809
ABSTRACT
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 highdimensional data lie on an intrinsically lowdimensional Riemannian manifold. The main idea is to formulate the dimensionality reduction problem as a classical problem in Riemannian geometry, i.e., 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 <em>k</em> 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 highdimensional data into a lowdimensional 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.
INDEX TERMS
Dimensionality reduction, manifold learning, manifold reconstruction, Riemannian manifolds, Riemannian normal coordinates.
CITATION
Tong Lin, Hongbin Zha, "Riemannian Manifold Learning", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.30, no. 5, pp. 796809, May 2008, doi:10.1109/TPAMI.2007.70735
REFERENCES
