Flag Manifolds in Modern Data Analysis

Abstract

We aim to outline the recent applications of flag manifolds for data analysis. Flags are nested sequences of subspaces of a vector space and the set of all flags with the same increasing subspace dimensions admits a Riemannian manifold structure called the flag manifold. The nested subspaces in a flag are parameterized by ordered orthonormal basis vectors collected into one matrix. Common algorithms like the eigenvalue decomposition extract eigenvectors that, when ordered by their eigenvalues, provide a flag (nested subspace) representation for the data. Using this perspective, flags have been identified as outputs from algorithms like principal component analysis, principal geodesic analysis, and subspace averages. Deeper investigation into the use of flags in data analysis has led to robust flag averages for computer vision and an improved formulation of probabilistic PCA called principal subspace analysis. Furthermore, flag structure has been embedded into many variants of robust subspace recovery methods through 'flagification' and a general lifting of subspace optimizations to flags is done through the 'flag trick'. Finally, flags are used as hierarchy-preserving data representations via a flag decomposition for hierarchical datasets. In this talk, we discuss the utility of flags for data analysis and offer concrete examples where the power of flags is used to improve motion averaging, outlier detection, dimensionality reduction, and more.