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I primarily work with non Euclidean data and am looking to extend concepts of 'variance' to Riemannian manifolds. I am aware of Karcher variance, but I need efficient ways to solve for it. For example, using Rank Revealing QR decomposition (RRQR)[1] provides some information regarding the sample variance for data points in vector spaces. I have also found a paper that uses this on manifolds[2].

I was wondering if there are other interpretations to the notion of variance that I can exploit? I guess in short - I want to know what are the different optimization problems one would solve for in vector spaces, that would give the sample variance as the solution.

Thanks.

[1] M. Gu and S. Eisenstat, Efficient algorithms for computing a strong rank-revealing QR factorization, SIAM Journal on Scientific Computing, 1996.

[2] N. Shroff, P. K. Turaga, and R. Chellappa. Manifold precis: An annealing technique for diverse sampling of manifolds. In NIPS, 2011.

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