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Principal component analysis (PCA), along with its extensions to manifoldsand outlier contaminated data, have been indispensable in computer vision andmachine learning. In this work, we present a unifying formalism for PCA and itsvariants, and introduce a framework based on the flags of linear subspaces, \iea hierarchy of nested linear subspaces of increasing dimension, which not onlyallows for a common implementation but also yields novel variants, not exploredpreviously. We begin by generalizing traditional PCA methods that eithermaximize variance or minimize reconstruction error. We expand theseinterpretations to develop a wide array of new dimensionality reductionalgorithms by accounting for outliers and the data manifold. To devise a commoncomputational approach, we recast robust and dual forms of PCA as optimizationproblems on flag manifolds. We then integrate tangent space approximations ofprincipal geodesic analysis (tangent-PCA) into this flag-based framework,creating novel robust and dual geodesic PCA variations. The remarkableflexibility offered by the 'flagification' introduced here enables even morealgorithmic variants identified by specific flag types. Last but not least, wepropose an effective convergent solver for these flag-formulations employingthe Stiefel manifold. Our empirical results on both real-world and syntheticscenarios, demonstrate the superiority of our novel algorithms, especially interms of robustness to outliers on manifolds.

Fun with Flags: Robust Principal Directions via Flag Manifolds