The Flag Manifold as a Tool for Analyzing and Comparing Sets of Data Sets

The shape and orientation of data clouds reflect variability in observations that can confound pattern recognition systems. Subspace methods, utilizing Grassmann manifolds, have been a great aid in dealing with such variability. However, this usefulness begins to falter when the data cloud contains sufficiently many outliers corresponding to stray elements from another class or when the number of data points is larger than the number of features. We illustrate how nested subspace methods, utilizing flag manifolds, can help to deal with such additional confounding factors. Flag manifolds, which are parameter spaces for nested sequences of subspaces, are a natural geometric generalization of Grassmann manifolds. We utilize and extend known algorithms for determining the minimal length geodesic, the initial direction generating the minimal length geodesic, and the distance between any pair of points on a flag manifold. The approach is illustrated in the context of (hyper) spectral imagery showing the impact of ambient dimension, sample dimension, and flag structure.