# A Flag Decomposition for Hierarchical Datasets

arXiv: https://arxiv.org/abs/2502.07782 · HTML: https://mlsanity.com/paper/2502.07782/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 59 references that are in the map.
- First posted on arXiv 11 Feb 2025.

## Builds on (most cited first, 18 of 18)

- [Prototypical Networks for Few-shot Learning](https://mlsanity.com/paper/1703.05175/) (2017)
- [EuroSAT: A Novel Dataset and Deep Learning Benchmark for Land Use and Land Cover Classification](https://mlsanity.com/paper/1709.00029/) (2017)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Fast, Robust and Non-convex Subspace Recovery](https://mlsanity.com/paper/1406.6145/) (2014)
- [Dual Principal Component Pursuit](https://mlsanity.com/paper/1510.04390/) (2015)
- [Optimization on flag manifolds](https://mlsanity.com/paper/1907.00949/) (2019)
- [A Scalable Second Order Method for Ill-Conditioned Matrix Completion from Few Samples](https://mlsanity.com/paper/2106.02119/) (2021)
- [Representing Part-Whole Hierarchies in Foundation Models by Learning Localizability, Composability, and Decomposability from Anatomy via Self-Supervision](https://mlsanity.com/paper/2404.15672/) (2024)
- [Iteratively Reweighted Least Squares for Basis Pursuit with Global Linear Convergence Rate](https://mlsanity.com/paper/2012.12250/) (2020)
- [The Flag Median and FlagIRLS](https://mlsanity.com/paper/2203.04437/) (2022)
- [Chordal Averaging on Flag Manifolds and Its Applications](https://mlsanity.com/paper/2303.13501/) (2023)
- [HyperSDFusion: Bridging Hierarchical Structures in Language and Geometry for Enhanced 3D Text2Shape Generation](https://mlsanity.com/paper/2403.00372/) (2024)
- [Beyond Low Rank + Sparse: Multi-scale Low Rank Matrix Decomposition](https://mlsanity.com/paper/1507.08751/) (2015)
- [Fun with Flags: Robust Principal Directions via Flag Manifolds](https://mlsanity.com/paper/2401.04071/) (2024)
- [The curse of isotropy: from principal components to principal subspaces](https://mlsanity.com/paper/2307.15348/) (2023)
- [Nested subspace learning with flags](https://mlsanity.com/paper/2502.06022/) (2025)
- [On a minimum enclosing ball of a collection of linear subspaces](https://mlsanity.com/paper/2003.12455/) (2020)

## Built on by (most cited first, 1 of 1)

- [Deep Hierarchical Learning with Nested Subspace Networks for Large Language Models](https://mlsanity.com/paper/2509.17874/) (2025)

## Abstract (arXiv)

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.