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Beyond R-barycenters: an effective averaging method on Stiefel and Grassmann manifolds

Florent Bouchard, Nils Laurent, Salem Said, Nicolas Le Bihan

TL;DR

The paper tackles averaging data on smooth manifolds, where the Riemannian Fréchet mean is often impractical due to cost or inavailability. It introduces RL-barycenters, which replace expensive inverse retractions with simple liftings, yielding projected arithmetic means on the Stiefel and Grassmann manifolds. The main contributions are closed-form RL-barycenters: for Stiefel, the projected mean corresponds to uf(mean) for polar projection and qf(mean) for QR projection, while for Grassmann, the mean is obtained by projecting the ambient mean onto Gr via top-k eigenvectors. Numerical results on simulated data show that these projected means are computationally cheaper and competitive with existing R-barycenters and close to the Riemannian mean, highlighting practical impact for signal processing and machine learning tasks. Code for reproducibility is provided at the authors' GitHub repository.

Abstract

In this paper, the issue of averaging data on a manifold is addressed. While the Fréchet mean resulting from Riemannian geometry appears ideal, it is unfortunately not always available and often computationally very expensive. To overcome this, R-barycenters have been proposed and successfully applied to Stiefel and Grassmann manifolds. However, R-barycenters still suffer severe limitations as they rely on iterative algorithms and complicated operators. We propose simpler, yet efficient, barycenters that we call RL-barycenters. We show that, in the setting relevant to most applications, our framework yields astonishingly simple barycenters: arithmetic means projected onto the manifold. We apply this approach to the Stiefel and Grassmann manifolds. On simulated data, our approach is competitive with respect to existing averaging methods, while computationally cheaper.

Beyond R-barycenters: an effective averaging method on Stiefel and Grassmann manifolds

TL;DR

The paper tackles averaging data on smooth manifolds, where the Riemannian Fréchet mean is often impractical due to cost or inavailability. It introduces RL-barycenters, which replace expensive inverse retractions with simple liftings, yielding projected arithmetic means on the Stiefel and Grassmann manifolds. The main contributions are closed-form RL-barycenters: for Stiefel, the projected mean corresponds to uf(mean) for polar projection and qf(mean) for QR projection, while for Grassmann, the mean is obtained by projecting the ambient mean onto Gr via top-k eigenvectors. Numerical results on simulated data show that these projected means are computationally cheaper and competitive with existing R-barycenters and close to the Riemannian mean, highlighting practical impact for signal processing and machine learning tasks. Code for reproducibility is provided at the authors' GitHub repository.

Abstract

In this paper, the issue of averaging data on a manifold is addressed. While the Fréchet mean resulting from Riemannian geometry appears ideal, it is unfortunately not always available and often computationally very expensive. To overcome this, R-barycenters have been proposed and successfully applied to Stiefel and Grassmann manifolds. However, R-barycenters still suffer severe limitations as they rely on iterative algorithms and complicated operators. We propose simpler, yet efficient, barycenters that we call RL-barycenters. We show that, in the setting relevant to most applications, our framework yields astonishingly simple barycenters: arithmetic means projected onto the manifold. We apply this approach to the Stiefel and Grassmann manifolds. On simulated data, our approach is competitive with respect to existing averaging methods, while computationally cheaper.
Paper Structure (9 sections, 2 theorems, 34 equations, 2 figures)

This paper contains 9 sections, 2 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Stiefel and Grassmann manifolds
  4. Barycenters on matrix manifolds
  5. Projection based barycenters
  6. Numerical experiments
  7. Conclusion and perspectives
  8. Projection on the Grassmann manifold
  9. $RL$-barycenter on Stiefel based on the QR decomposition

Key Result

Proposition 1

Given the retraction eq:retr_proj and the lifting eq:lift_proj, the $RL$-barycenter of $\{\boldsymbol{M}_i\}_{i=1}^{n}$, according to Definition def:RLbarycenter, is

Figures (2)

  • Figure 1: Medians (solid lines), 10% and 90% quantiles (filled areas) over $100$ realizations of error measure \ref{['eq:err_st']} of mean estimators on the Stiefel manifold $\textup{St}_{p,k}$. "$R$ polar" and "$R$ QR" correspond to $R$-barycenters with polar and QR retractions. "proj polar" and "proj QR" correspond to the projected arithmetic means with the projections on $\textup{St}_{p,k}$ based on the polar and QR decompositions, respectively. In these simulations, $p=10$ and $k=5$.
  • Figure 2: Medians (solid lines), 10% and 90% quantiles (filled areas) over $100$ realizations of error measure \ref{['eq:err_gr']} of mean estimators on the Grassmann manifold $\textup{Gr}_{p,k}$. In the legend, "proj evd" corresponds to the projected arithmetic mean with the projection on $\textup{Gr}_{p,k}$ based on the eigenvalue decomposition. In these simulations, $p=10$, $k=5$ and $\sigma=0.5$.

Theorems & Definitions (5)

  • Definition 1: $RL$-barycenters
  • Proposition 1: Projection based barycenters
  • proof
  • Proposition 2: Projection on the Grassmann manifold
  • proof