Computing medians and means in Hadamard spaces
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Submission date: 01. Feb. 2013 (revised version: February 2013)
published in: SIAM journal on optimization, 24 (2014) 3, p. 1542-1566
DOI number (of the published article): 10.1137/140953393
MSC-Numbers: 51F99, 60B99, 92B05, 49M27, 62E99
Keywords and phrases: Hadamard space, tree space, Frechet mean, law of large numbers, proximal point algorithm, median and mean
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The geometric median as well as the Frechet mean of points in an Hadamard space are important in both theory and applications. Surprisingly, no algorithms for their computation are hitherto known. To address this issue, we use a split version of the proximal point algorithm for minimizing a sum of convex functions and prove that this algorithm produces a sequence converging to a minimizer of the objective function, which extends a recent result of D. Bertsekas (2001) into Hadamard spaces. The method is quite robust and not only does it yield algorithms for the median and the mean, but it also applies to various other optimization problems. We moreover show that another algorithm for computing the Frechet mean can be derived from the law of large numbers due to K.-T. Sturm (2002). In applications, computing medians and means is probably most needed in tree space, which is an instance of an Hadamard space, invented by Billera, Holmes, and Vogtmann (2001) as a tool for averaging phylogenetic trees. It turns out, however, that it can be also used to model numerous other tree-like structures. Since there now exists a polynomial-time algorithm for computing geodesics in tree space due to M. Owen and S. Provan (2011), we obtain ecient algorithms for computing medians and means, which can be directly used in practice.