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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.
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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
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This is outside of my field, and there are no sources for the terminology of the article; so it is hard for me to know what is the "correct" terminology. All I can say, though, is that the lack of sources and the fact that the terminology is not explained and differs from a terminology that seems to
345:
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)
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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.
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Fréchet variance" to refer to what would correspond — using the terminology of this
Knowledge article — to the Fréchet variance with respect to the Fréchet mean. A quick Google search shows that this use of the terms is not
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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.
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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.
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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?
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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
691:"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. --
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Article says "the square root of the
Euclidean distance, i.e. the taxicab distance." Sounds false to me. Should that be removed?--
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If there were a substantial body of literature that uses the phrase "Karcher mean", then what would
Knowledge policy dictate?
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757:? The definitions of mean and variance there seem to match the ones given in this article. Please clarify.
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Krakowski et al. (2007), On the Computation of the Karcher Mean on Spheres and Special Orthogonal Groups
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Knowledge falsely writes: Karcher means are a closely related construction named after Hermann Karcher.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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is correct: the definition of the means match; but what is referred to as "Fréchet variance
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The root problem with this statement is that the article earlier specifies
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The definitions seem in contrast to definitions from the litterature
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Hartley et al. (2010), Rotation averaging and weak convexity
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is the Euclidean distance. This is correct. Restricting to
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Riemannian Center of Mass and so called karcher mean,
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775:: No, @
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