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) provides some information regarding the sample variance for data points in vector spaces. I have also found a paper that uses this on manifolds.
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.
 M. Gu and S. Eisenstat, Efﬁcient algorithms for computing a strong rank-revealing QR factorization, SIAM Journal on Scientiﬁc Computing, 1996.
 N. Shroff, P. K. Turaga, and R. Chellappa. Manifold precis: An annealing technique for diverse sampling of manifolds. In NIPS, 2011.