[Paper Review] A novel algorithm for computing the Frechet mean in Hadamard spaces

Miroslav Bačák|arXiv (Cornell University)|Oct 8, 2012

Morphological variations and asymmetryMathematics7 citations

TL;DR

This paper presents the first known algorithms for computing the Frechet mean and geometric median in Hadamard spaces, using a split proximal point algorithm and a law of large numbers approach. It establishes convergence in Hadamard spaces and enables efficient computation of medians and means, particularly in tree space, with polynomial-time geodesic computation enabling practical use.

ABSTRACT

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 efficient algorithms for computing medians and means, which can be directly used in practice.

Motivation & Objective

• To develop the first computationally feasible algorithms for computing the Frechet mean and geometric median in Hadamard spaces.
• To extend D. Bertsekas' (2001) proximal point convergence result to the setting of Hadamard spaces.
• To enable practical computation of medians and means in tree space, a key application domain for phylogenetic tree averaging.
• To demonstrate the robustness and broad applicability of the proposed method to various optimization problems beyond mean and median computation.

Proposed method

• A split version of the proximal point algorithm is used to minimize a sum of convex functions in Hadamard spaces.
• Convergence of the algorithm to a minimizer of the objective function is proven, extending Bertsekas' result to Hadamard spaces.
• The method is applied to compute the Frechet mean and geometric median by formulating them as minimization problems.
• An alternative algorithm for the Frechet mean is derived from K.-T. Sturm's (2002) law of large numbers in Hadamard spaces.
• The approach leverages the existence of a polynomial-time algorithm for computing geodesics in tree space.
• The combination of the proximal point method and efficient geodesic computation yields practical, efficient algorithms for real-world applications.

Experimental results

Research questions

• RQ1 Can a convergent algorithm be constructed for computing the Frechet mean in Hadamard spaces, where no such algorithms were previously known?
• RQ2 How can the proximal point algorithm be adapted to ensure convergence in the general setting of Hadamard spaces?
• RQ3 Can the law of large numbers in Hadamard spaces be used to derive a practical algorithm for computing the Frechet mean?
• RQ4 To what extent is the proposed method robust and applicable to other optimization problems beyond mean and median computation?
• RQ5 How efficiently can the Frechet mean and median be computed in tree space, given the availability of polynomial-time geodesic computation?

Key findings

• The split proximal point algorithm converges to a minimizer of the objective function in Hadamard spaces, extending Bertsekas' (2001) result to this non-Euclidean setting.
• The algorithm enables the first practical computation of the Frechet mean and geometric median in Hadamard spaces.
• The method is robust and generalizable to a wide range of optimization problems in Hadamard spaces.
• An alternative algorithm for the Frechet mean is derived from Sturm's (2002) law of large numbers, providing a second computational pathway.
• In tree space, the combination of the proposed algorithms and polynomial-time geodesic computation yields efficient, directly usable algorithms for median and mean computation.
• The results establish a foundation for computing medians and means in tree-like structures, with immediate applications in phylogenetics and other domains.