Talk:Fréchet mean

From Hermann Karcher's webpage:

CORRECTION

Wikipedia falsely writes: Karcher means are a closely related construction named after Hermann Karcher.

True is: Karcher mean is the renaming of the Riemannian Center of Mass construction developed by Karsten Grove and Hermann Karcher in: How to conjugate C1-close group actions, Math.Z. 132, 1973, pp 11-20.

My 1977 paper with Riemannian Center of Mass in the title is more easily found by google. But that does not justify such a renaming. There I also quote: Grove, K., Karcher, H., Ruh, E. A., Jacobi fields and Finsler metrics on compact Lie Groups ..., Math. Ann. 211, 1974, pp. 7-21, where the center is defined on SO(n) on much larger sets than can be done with their Riemannian metric. In Buser, P., Karcher, H., Gromov's Almost Flat Manifolds, Soc. Mat. France, Astérisque 81, 1981, the center is defined on nilpotent Lie groups just using their connection, as in the Euclidean affine case. On spheres the squared distance does not work so well since its Hessian has different eigenvalues in radial and tangential direction. It is easier to use 1- cos(d(.,p)) instead. In Chern's book Global Differential Geometry, MAA Studies in Mathematics, Vol 27, 1989, my article Riemannian Comparison Constructions explains about such modified distance functions. The book is out of print, but google finds my contribution on my Homepage.

See also:

Hermann Karcher, Riemannian Center of Mass and so called karcher mean, arXiv:1407.2087, 3 July 2014.

Typometer (talk) 14:24, 1 June 2014 (UTC)

One can definitely find published papers that refer to the minimum (local or global) as the "Karcher mean". They all seem to cite Karcher's 1977 paper, which calls it the "Riemannian center of mass" as above. Here are two examples.

• Hartley et al. (2010), Rotation averaging and weak convexity
• Krakowski et al. (2007), On the Computation of the Karcher Mean on Spheres and Special Orthogonal Groups

If there were a substantial body of literature that uses the phrase "Karcher mean", then what would Wikipedia policy dictate? Mgnbar (talk) 00:14, 25 August 2014 (UTC)

Taxicab distance falsely mentioned?

Article says "the square root of the Euclidean distance, i.e. the taxicab distance." Sounds false to me. Should that be removed?--אדי פ' (talk) 23:51, 2 May 2020 (UTC)

I was also worried about that edit. I think that it's trying to get at the following idea. Given points x1, ..., xn in the real number line, the mean is the value of x that minimizes SUM |xi - x|^2, while the median is the value of x that minimizes SUM |xi - x|. So the mean is the Euclidean (L^2) Frechet mean and the median is the taxicab (L^1) Frechet mean. The L^1 distance is the square root of the Euclidean distance, although that part could be worded much more clearly. Is all of that right? Mgnbar (talk) 12:54, 3 May 2020 (UTC)

The root problem with this statement is that the article earlier specifies $\Psi (p)=\sum _{i}d^{2}(p,x_{i})$ for a distance function $d$, but the reference for the statement uses a different definition $\Psi (p)=\sum _{i}d^{\alpha }(p,x_{i})$, noting that $\alpha =2$ produces the mean and $\alpha =1$ produces the median when $d$ is the Euclidean distance. This is correct. Restricting to $\alpha =2$ as this article previously does excludes the median, since the square root of the euclidean distance is not a distance function. To be rigorous, the article either needs to generalize to $\alpha \neq 2$, or, if it wants to keep $\alpha =2$, remove this claim.

"the square root of the Euclidean distance, i.e. the taxicab distance." though is wrong, no matter what. The square root of the Euclidean distance is not the taxicab distance at all. --73.13.145.177 (talk) 02:37, 18 August 2020 (UTC)

You've clarified it well. I agree. Mgnbar (talk) 12:23, 18 August 2020 (UTC)

The definitions seem in contrast to definitions from the litterature

The article "Frèchet Analysis of Variance for Random Objects" has another definition of the means and variance. I cannot find information supporting the formulation in the article from the source either. Sprint99 (talk) 21:11, 16 September 2023 (UTC)

Do you mean this article? The definitions of mean and variance there seem to match the ones given in this article. Please clarify. Mgnbar (talk) 15:40, 17 September 2023 (UTC)