# Fun with Flags: Robust Principal Directions via Flag Manifolds

arXiv: https://arxiv.org/abs/2401.04071 · HTML: https://mlsanity.com/paper/2401.04071/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 83 references that are in the map.
- First posted on arXiv 8 Jan 2024.

## Builds on (most cited first, 19 of 19)

- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices](https://mlsanity.com/paper/1407.1120/) (2014)
- [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)
- [Optimal Algorithms for L1-subspace Signal Processing](https://mlsanity.com/paper/1405.6785/) (2014)
- [Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design](https://mlsanity.com/paper/2112.03402/) (2021)
- [Robust PCA by Manifold Optimization](https://mlsanity.com/paper/1708.00257/) (2017)
- [Optimization on flag manifolds](https://mlsanity.com/paper/1907.00949/) (2019)
- [Riemannian conjugate gradient methods: General framework and specific algorithms with convergence analyses](https://mlsanity.com/paper/2112.02572/) (2021)
- [Efficient L1-Norm Principal-Component Analysis via Bit Flipping](https://mlsanity.com/paper/1610.01959/) (2016)
- [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)
- [Kernel Multivariate Analysis Framework for Supervised Subspace Learning: A Tutorial on Linear and Kernel Multivariate Methods](https://mlsanity.com/paper/1310.5089/) (2013)
- [Nonlinearities and Adaptation of Color Vision from Sequential Principal Curves Analysis](https://mlsanity.com/paper/1602.00236/) (2016)
- [Camera Pose Filtering with Local Regression Geodesics on the Riemannian Manifold of Dual Quaternions](https://mlsanity.com/paper/1704.07072/) (2017)
- [Closed-form geodesics and trust-region method to calculate Riemannian logarithms on Stiefel and its quotient manifolds](https://mlsanity.com/paper/2103.13327/) (2021)
- [Principal Component Analysis in Space Forms](https://mlsanity.com/paper/2301.02750/) (2023)
- [Principal nested shape space analysis of molecular dynamics data](https://mlsanity.com/paper/1903.09445/) (2019)
- [An Extension of Fast Iterative Shrinkage-thresholding to Riemannian Optimization for Sparse Principal Component Analysis](https://mlsanity.com/paper/1909.05485/) (2019)

## Built on by (most cited first, 5 of 5)

- [MRFD: Multi-Region Fusion Decoding with Self-Consistency for Mitigating Hallucinations in LVLMs](https://mlsanity.com/paper/2508.10264/) (2025)
- [Deep Hierarchical Learning with Nested Subspace Networks for Large Language Models](https://mlsanity.com/paper/2509.17874/) (2025)
- [Nested subspace learning with flags](https://mlsanity.com/paper/2502.06022/) (2025)
- [Parallel transport on matrix manifolds and Exponential Action](https://mlsanity.com/paper/2408.06054/) (2024)
- [A Flag Decomposition for Hierarchical Datasets](https://mlsanity.com/paper/2502.07782/) (2025)

## Abstract (arXiv)

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.