Computing Medians and Means in Hadamard Spaces

Abstract

Abstract. The geometric median as well as the Fréchet 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 Fréchet 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 efficient algorithms for computing medians and means, which can be directly used in practice.

Discrete-time gradient flows and law of large numbers in Alexandrov spaces

“…On the one hand, this result generalizes the ones given for Euclidean spaces in [9,8,32,33] and for NPC spaces in [6]. On the other hand, this is also a generalization of the "no dice" approximation result given in NPC spaces for the barycenter, which is the minimizer of f (x) = n i=1 w i d(x, a i ) 2 with fixed points a i ∈ X, in [28,16].…”

“…In fact, it is known that in the infinite dimensional case we cannot always have convergence under these assumptions [6]. However, if we assume that f is K-convex for positive K, then the assumption of local compactness can be dropped.…”

Polyhedral computational geometry for averaging metric phylogenetic trees

“…One min cut with respect to the flow in Fig. 3(b) has H = {x 1 , x 2 , x 3 , t 1 , t } and H its complement; this corresponds to the pairs 4 , t 5 , t 6 , t 7 }, with squared ratios 9 9 = 14 14 .…”

“…Our decision to use this definition is motivated by work of Sturm [32], who identified the Fréchet mean as a theoretically rich statistical object associated with sampling from a specified distribution on a global NPC space (see Theorem 2.4). Fréchet means in tree space and the algorithm for computing them that arises from Sturm's work (Algorithm 2.5) have been independently developed by Bačák [4].…”

The modified proximal point algorithm in CAT(0) spaces

“…Recently, Bačák [4] employed a split version of the PPA for minimizing a sum of convex functions in complete CAT(0) spaces. Other interesting results can be also found in [9,18].…”