  Principal component analysis (PCA), along with its extensions to manifolds
and outlier contaminated data, have been indispensable in computer vision and
machine learning. In this work, we present a unifying formalism for PCA and its
variants, and introduce a framework based on the flags of linear subspaces, ie
a hierarchy of nested linear subspaces of increasing dimension, which not only
allows for a common implementation but also yields novel variants, not explored
previously. We begin by generalizing traditional PCA methods that either
maximize variance or minimize reconstruction error. We expand these
interpretations to develop a wide array of new dimensionality reduction
algorithms by accounting for outliers and the data manifold. To devise a common
computational approach, we recast robust and dual forms of PCA as optimization
problems on flag manifolds. We then integrate tangent space approximations of
principal geodesic analysis (tangent-PCA) into this flag-based framework,
creating novel robust and dual geodesic PCA variations. The remarkable
flexibility offered by the 'flagification' introduced here enables even more
algorithmic variants identified by specific flag types. Last but not least, we
propose an effective convergent solver for these flag-formulations employing
the Stiefel manifold. Our empirical results on both real-world and synthetic
scenarios, demonstrate the superiority of our novel algorithms, especially in
terms of robustness to outliers on manifolds.
