The flag manifold as a tool for analyzing and comparing data sets

TL;DR

This paper introduces the use of flag manifolds, a geometric tool for analyzing nested subspaces, to improve data set comparison and robustness against outliers in pattern recognition tasks.

Contribution

It proposes algorithms for computing distances and geodesics on flag manifolds, extending subspace methods to handle complex data variability and outliers.

Findings

Flag manifolds enhance data comparison in high-dimensional spaces.

Algorithms effectively measure distances and directions on flag manifolds.

Application to spectral imagery demonstrates practical benefits.

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.