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

Abstract

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 subspaces, are a natural geometric generalization of Grassmann manifolds. To make practical comparisons on a flag manifold, algorithms are proposed for determining the distances between points [A], [B] on a flag manifold, where A and B are arbitrary orthogonal matrix representatives for [A] and [B], and for determining the initial direction of these minimal length geodesics. The approach is illustrated in the context of (hyper) spectral imagery showing the impact of ambient dimension, sample dimension, and flag structure.

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“…If the object is Lambertian then this cone has been shown to lie close to a low dimensional linear space which can in turn be represented by a point on a Grassmannian (Beveridge et al, 2008). The flag manifold comes equipped with geometric features capable of representing sets of data where the number of points is larger than the number of dimensions in the ambient space (Ma et al, 2021). We note that the flag mean proposed in Marrinan et al (2014Marrinan et al ( , 2015 and Mankovich et al (2022), and its various extensions, are a special case of the work proposed here.…”