  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.
