Fréchet Mean (Karcher Flow)

The Riemannian generalization of the arithmetic mean: the point minimizing the sum of squared geodesic distances to a set of input points.

Algorithm Summary

Definition

Karcher flow gradient step:

Repeat until is below tolerance.

FrechetConfig

pub struct FrechetConfig {
    pub max_iters:  usize,  // default: 200
    pub tol:        f64,    // default: 1e-8 - stop when gradient norm < tol
    pub step_size:  f64,    // default: 1.0  - Karcher step size (1.0 = exact gradient flow)
}

Code Examples

Mean of points on S²

use cartan::prelude::*;
use cartan::manifolds::Sphere;
use cartan_optim::{frechet_mean, FrechetConfig};
 
let s2 = Sphere::<3>;
let mut rng = rand::rng();
 
let points: Vec<_> = (0..20).map(|_| s2.random_point(&mut rng)).collect();
 
// init = None → uses points[0] as starting estimate
let mean = frechet_mean(&s2, &points, None, &FrechetConfig::default());
assert!(mean.converged);

Custom initial estimate

// Provide a custom starting point for the Karcher iteration
let init = s2.random_point(&mut rng);
let mean = frechet_mean(&s2, &points, Some(init), &FrechetConfig::default());

Numerical Notes

Convergence radius: Karcher flow converges to the global minimum only when the points are contained within a geodesic ball of radius less than . Outside this radius there may be multiple local minima.

Initialization: frechet_mean initializes at the first input point when None is passed. For widely spread data, consider running from multiple initializations.

Negative curvature: on Cartan-Hadamard manifolds (SPD with affine-invariant metric, ), the Fréchet mean is unique and Karcher flow converges globally.

Compared to minimize_rgd: use frechet_mean (not RGD on the sum-of-squares objective; it exploits the special structure of the objective and is numerically more stable.

Applications

• Averaging rotation matrices: SO(3) mean for sensor fusion
• Covariance averaging: SPD mean for brain-computer interfaces / fMRI
• Directional statistics: mean of unit vectors on
• Point cloud alignment: Fréchet mean of SE(3) poses in SLAM