# Chordal Averaging on Flag Manifolds and Its Applications

arXiv: https://arxiv.org/abs/2303.13501 · HTML: https://mlsanity.com/paper/2303.13501/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 70 references that are in the map.
- First posted on arXiv 23 Mar 2023.

## Builds on (most cited first, 17 of 17)

- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Shonan Rotation Averaging: Global Optimality by Surfing SO(p)n](https://mlsanity.com/paper/2008.02737/) (2020)
- [Quaternion Equivariant Capsule Networks for 3D Point Clouds](https://mlsanity.com/paper/1912.12098/) (2019)
- [Learning Transformation Synchronization](https://mlsanity.com/paper/1901.09458/) (2019)
- [Probabilistic Permutation Synchronization using the Riemannian Structure of the Birkhoff Polytope](https://mlsanity.com/paper/1904.05814/) (2019)
- [Bayesian Pose Graph Optimization via Bingham Distributions and Tempered Geodesic MCMC](https://mlsanity.com/paper/1805.12279/) (2018)
- [Grassmannian Learning: Embedding Geometry Awareness in Shallow and Deep Learning](https://mlsanity.com/paper/1808.02229/) (2018)
- [Synchronizing Probability Measures on Rotations via Optimal Transport](https://mlsanity.com/paper/2004.00663/) (2020)
- [Defining the Pose of any 3D Rigid Object and an Associated Distance](https://mlsanity.com/paper/1612.04631/) (2016)
- [Optimization on flag manifolds](https://mlsanity.com/paper/1907.00949/) (2019)
- [Robust Single Rotation Averaging](https://mlsanity.com/paper/2004.00732/) (2020)
- [The Flag Median and FlagIRLS](https://mlsanity.com/paper/2203.04437/) (2022)
- [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)
- [Intrinsic Grassmann Averages for Online Linear, Robust and Nonlinear Subspace Learning](https://mlsanity.com/paper/1702.01005/) (2017)
- [Hybrid Camera Pose Estimation with Online Partitioning for SLAM](https://mlsanity.com/paper/1908.01797/) (2019)
- [Synchronization over Cartan motion groups via contraction](https://mlsanity.com/paper/1612.00059/) (2016)

## Built on by (most cited first, 9 of 9)

- [Robust Single Rotation Averaging Revisited](https://mlsanity.com/paper/2309.05388/) (2023)
- [Fun with Flags: Robust Principal Directions via Flag Manifolds](https://mlsanity.com/paper/2401.04071/) (2024)
- [Frame Representation Hypothesis: Multi-Token LLM Interpretability and Concept-Guided Text Generation](https://mlsanity.com/paper/2412.07334/) (2024)
- [Forecasting Continuous Non-Conservative Dynamical Systems in SO(3)](https://mlsanity.com/paper/2508.07775/) (2025)
- [Nested subspace learning with flags](https://mlsanity.com/paper/2502.06022/) (2025)
- [A Flag Decomposition for Hierarchical Datasets](https://mlsanity.com/paper/2502.07782/) (2025)
- [Robustness of Generalized Median Computation for Consensus Learning in Arbitrary Spaces](https://mlsanity.com/paper/2503.05215/) (2025)
- [Wireless Dataset Similarity: Measuring Distances in Supervised and Unsupervised Machine Learning](https://mlsanity.com/paper/2601.01023/) (2026)
- [SNAP: A Self-Consistent Agreement Principle with Application to Robust Computation](https://mlsanity.com/paper/2602.02013/) (2026)

## Abstract (arXiv)

This paper presents a new, provably-convergent algorithm for computing the flag-mean and flag-median of a set of points on a flag manifold under the chordal metric. The flag manifold is a mathematical space consisting of flags, which are sequences of nested subspaces of a vector space that increase in dimension. The flag manifold is a superset of a wide range of known matrix spaces, including Stiefel and Grassmanians, making it a general object that is useful in a wide variety computer vision problems. To tackle the challenge of computing first order flag statistics, we first transform the problem into one that involves auxiliary variables constrained to the Stiefel manifold. The Stiefel manifold is a space of orthogonal frames, and leveraging the numerical stability and efficiency of Stiefel-manifold optimization enables us to compute the flag-mean effectively. Through a series of experiments, we show the competence of our method in Grassmann and rotation averaging, as well as principal component analysis. We release our source code under https://github.com/nmank/FlagAveraging.