Self-organizing mappings on the Grassmannian with applications to data analysis in high dimensions

摘要

We propose a method for extending Kohonen's self-organizing mapping to the geometric framework of the Grassmannian. The resulting algorithm serves as a prototype of the extension of the SOM to the setting of abstract manifolds. The ingredients required for this are a means to measure distance between two points, and a method to move one point in the direction of another. In practice, the data are not required to have a representation in Euclidean space. We discuss in detail how a point on a Grassmannian is moved in the direction of another along a geodesic path. We demonstrate the implementation of the algorithm on several illustrative data sets, hyperspectral images and gene expression data sets.