A Flag Decomposition for Hierarchical Datasets

TL;DR

This paper introduces a novel flag-based decomposition method for hierarchical datasets, enabling better preservation of data structure for applications like denoising, clustering, and few-shot learning.

Contribution

It presents a general algorithm to decompose hierarchical data into flag representations, extending beyond traditional matrix decompositions.

Findings

Effective in denoising hierarchical data

Improves clustering accuracy on complex datasets

Facilitates few-shot learning tasks

Abstract

Flag manifolds encode nested sequences of subspaces and serve as powerful structures for various computer vision and machine learning applications. Despite their utility in tasks such as dimensionality reduction, motion averaging, and subspace clustering, current applications are often restricted to extracting flags using common matrix decomposition methods like the singular value decomposition. Here, we address the need for a general algorithm to factorize and work with hierarchical datasets. In particular, we propose a novel, flag-based method that decomposes arbitrary hierarchical real-valued data into a hierarchy-preserving flag representation in Stiefel coordinates. Our work harnesses the potential of flag manifolds in applications including denoising, clustering, and few-shot learning.