Algorithms on Flag Manifolds for Knowledge Discovery in N-way Arrays
Colorado State University Fort Collins
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We proposed an approach for hyperspectral imagery classification that exploits the geometric framework of the Grassmannmanifold i.e., a parameterization of k dimensional subspaces of n-dimnsional space. The algorithm is particularly well suited to applications where sets of pixels are to be classified. Multiple pixels from a data class characterize the variability of the class information using a subspace representation. We use two metrics defined on the Grassmannian, chordal and geodesic, and one pseudometric, to compute pairwise distances between the points--subspaces. Once a distance matrix is generated, we use the classical multidimensional scaling to find a configuration of points with preserved or approximated original distances, thus realizing an embedding of the Grassmannian into Euclidean space. A sparse support vector machine SSVM trained in the embedding space simultaneously classifies embedded subspaces and selects a subset of optimal dimensions of the embedding for subsequentmodel reduction and data visualization.
- Numerical Mathematics