Visualizing data sets on the Grassmannian using self-organizing mappings

Abstract

We extend the self-organizing mapping algorithm to the problem of visualizing data on Grassmann manifolds. In this setting, a collection of k points in n-dimensions is represented by a k-dimensional subspace, e.g., via the singular value or QR-decompositions. Data assembled in this way is challenging to visualize given abstract points on the Grassmannian do not reside in Euclidean space. The extension of the SOM algorithm to this geometric setting only requires that distances between two points can be measured and that any given point can be moved towards a presented pattern. The similarity between two points on the Grassmannian is measured in terms of the principal angles between subspaces, e.g., the chordal distance. Further, we employ a formula for moving one subspace towards another along the shortest path, i.e., the geodesic between two points on the Grassmannian. This enables a faithful implementation of the SOM approach for visualizing data consisting of k-dimensional subspaces of n-dimensional Euclidean space. We illustrate the resulting algorithm on a hyperspectral imaging application.