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.

主成分分析（PCA）及其在计算机视观和机器学习中的扩展，通过线性子空间的标志引入了一个统一的形式，将传统的PCA方法推广为考虑异常值和数据流形的新的降维算法，并提出了一种基于标志流形的优化问题求解方法，通过Stiefel流形实现了收敛性的求解器。

旗帜玩乐：通过旗帜流形获得鲁棒的主方向