Tropical Fréchet Means: a polyhedral approach to exact optimization

Abstract

The Fréchet mean is a fundamental notion of central tendency defined as a minimizer of a sum of squared distances in a general metric space. In this paper, we study Fréchet means in tropical geometry -- a piecewise linear, combinatorial, and polyhedral variant of algebraic geometry -- by formulating and solving the associated tropical quadratic optimization problem. We give a geometric characterization of the collection of all tropical Fréchet means as a bounded set that is simultaneously tropically and classically convex, hence a polytrope. We establish the existence of positivity certificates for maxima of finitely many quadratic polynomials in $\mathbb{R}[x_1,\ldots,x_n]$ whose homogeneous quadratic components are sums of squares, which provides a symbolic framework for exact optimization. Using this structure, we develop algorithms for computing tropical Fréchet means and the associated Fréchet mean polytrope. We further describe a combinatorial type decomposition of the objective function induced by braid arrangements, yielding a piecewise quadratic representation and a fully symbolic method for exact computation.

Figures (8)

• Figure 1: The polytrope from \ref{['ex:polytrope']}. The three black dots are the tropical vertices while there are two additional pseudovertices
• Figure 2: This figure shows the intersection of balls centered at ${P = \{(0,0,8), (0,2,4), (0,5,3), (0,10,2)\}}$. The associated Fréchet mean polytrope $\overline{P}$ is the line segment from $(0,3,3)$ and $(0,4,4)$ and the minimal sum of squares is 34.
• Figure 3: The tropical Fréchet mean is an intersection of tropical spheres. We visualise points in $\mathbb{R}^3/\mathbb{R}\mathbf{1}$ by taking the representative with zero as its first coordinate. The figure shows the unique Fréchet mean $(0,0,-1)$ of the points from Example \ref{['ex:quadratic-optimisation']}.
• Figure 4: Tropical skinny triangle in the sense of Alexandrov. The correct Fréchet mean is shown in green, whereas Sturm's algorithm converges to a wrong point (in red).
• Figure 5: Braid arrangement in the tropical affine plane $\mathbb{R}^{2}/\mathbb{R}\mathbf{1}$. The chambers are marked with their covectors and types.

Theorems & Definitions (47)

• Definition 1
• Example 2
• Proposition 3: Joswig:ETC
• Example 4
• Theorem 5: DS:2004
• Corollary 6
• proof
• Lemma 7
• proof
• Definition 8