A Flag Decomposition for Hierarchical Datasets

2025-02-11

AI Summary

Traditional matrix factorization methods (e.g., SVD) fail to preserve hierarchical structure when modeling hierarchical data. To address this, we propose the first end-to-end flag manifold decomposition framework. Our method maps arbitrary real-valued hierarchical data to flag representations under Stiefel coordinates, explicitly encoding hierarchical constraints during dimensionality reduction. Leveraging differential-geometric gradient optimization, we jointly solve for hierarchy-aware orthogonal subspace decomposition on both the flag and Stiefel manifolds. Empirically, our approach significantly outperforms SVD and other baselines in denoising, subspace clustering, and few-shot learning—demonstrating the flag representation’s superior capacity for modeling hierarchical semantics. Our core contributions are threefold: (i) the first differentiable mapping from hierarchical data to the flag manifold; (ii) a geometrically consistent, hierarchy-faithful decomposition paradigm; and (iii) a task-generalizable framework enabling end-to-end learning across diverse downstream applications.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Representation Learning for Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Flag manifolds encode hierarchical 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.

Problem

• Decompose hierarchical datasets efficiently
• Preserve hierarchy in flag representation
• Enhance applications like denoising and clustering

Innovation

• Flag-based decomposition method
• Hierarchy-preserving flag representation
• Stiefel coordinates utilization