旗帜之趣：通过旗流形实现鲁棒主方向
Fun with Flags: Robust Principal Directions via Flag Manifolds
AI总结：
本文提出基于线性子空间旗流形的统一PCA框架，将鲁棒、对偶及tangent-PCA形式化为旗上优化，并给出收敛求解器，提升流形数据上的离群点鲁棒性。
摘要：
主成分分析（PCA）及其向流形和受离群点污染数据的扩展，在计算机视觉和机器学习中一直不可或缺。在本文中，我们提出了一种统一PCA及其变体的形式化方法，并引入了一个基于线性子空间旗（flags）的框架，即维度递增的嵌套线性子空间层次结构；该框架不仅支持通用实现，还产生了此前未曾探索的新变体。我们首先推广了传统PCA方法，这些方法要么最大化方差，要么最小化重构误差。我们通过考虑离群点和数据流形，扩展这些解释，以开发一系列新的降维算法。为设计通用计算方法，我们将PCA的鲁棒形式和对偶形式重新表述为旗流形上的优化问题。随后，我们将主测地线分析（principal geodesic analysis，tangent-PCA）的切空间近似整合到这个基于旗的框架中，创造出新颖的鲁棒和对偶测地线PCA变体。本文提出的“旗化（flagification）”所提供的显著灵活性，甚至支持由特定旗类型标识的更多算法变体。最后同样重要的是，我们利用Stiefel流形为这些旗形式化问题提出了一种有效的收敛求解器。我们在真实世界和合成场景上的实验结果证明了新算法的优越性，尤其是在流形上对离群点的鲁棒性方面。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
