Robust Representation and Estimation of Barycenters and Modes of Probability Measures on Metric Spaces

Abstract

This paper is concerned with the problem of defining and estimating statistics for distributions on spaces such as Riemannian manifolds and more general metric spaces. The challenge comes, in part, from the fact that statistics such as means and modes may be unstable: for example, a small perturbation to a distribution can lead to a large change in Fréchet means on spaces as simple as a circle. We address this issue by introducing a new merge tree representation of barycenters called the barycentric merge tree (BMT), which takes the form of a measured metric graph and summarizes features of the distribution in a multiscale manner. Modes are treated as special cases of barycenters through diffusion distances. In contrast to the properties of classical means and modes, we prove that BMTs are stable -- this is quantified as a Lipschitz estimate involving optimal transport metrics. This stability allows us to derive a consistency result for approximating BMTs from empirical measures, with explicit convergence rates. We also give a provably accurate method for discretely approximating the BMT construction and use this to provide numerical examples for distributions on spheres and shape spaces.

Figures (8)

• Figure 1: Barycentric merge tree example. Left: A probability density on the circle. The points marked by $\star$ are local minima of $\sigma_2$, with the darkest indicating the (global) Fréchet mean. Right: The barycentric merge tree summarizes the local minima, together with the shape of deviation function.
• Figure 2: Instability of global minima of $\sigma_p$ on the circle. (a) A pdf on the circle, its corresponding deviation function $\sigma_2$, and the global minimizers of $\sigma_2$. The deviation function and the pdf are both rescaled in this plot for visual clarity. (b) Four draws of 100 samples from the distribution shown in (a); $\sigma_2$ is plotted for each empirical distribution, along with its global minimizer, which varies wildly between samples. (c) A distribution on the circle consisting of equally weighted Dirac masses supported on antipodal points (represented by equal length arrows) and its deviation function $\sigma_1$. Here, $\sigma_1$ is constant, so every point on the circle is a minimizer (i.e., a median). (d) Measures consisting of Dirac masses where the weight on the mass at $(1,0)$ is given by $\mu_1$, together with their associated deviation functions $\sigma_1$. If $\mu_1 > 1/2$, the unique median lies at $(1,0)$, while the median lies at $(-1,0)$ if $\mu_1 < 1/2$, indicated with a star in either case.
• Figure 3: Stability of Barycentric Merge Trees on the circle. (a) A circle with a bimodal distribution, its deviation function $\sigma_2$ and the Fréchet mean set, as in Figure \ref{['fig:FrechetMeansCircle']}. (b) The BMT for the distribution in (a); the leaves of the tree correspond to the global minima of $\sigma_2$ (indicated by the stars), and the height of the merge point at the top corresponds to the maximum value of $\sigma_2$. (c) BMTs for empirical distributions drawn from the pdf. Leaves of the trees correspond to marked points on the circles. Observe that the merge trees all have similar structure, reflecting the stability of the BMTs as summaries of the behavior of the deviation function.
• Figure 4: Fréchet variance functions associated with the heat kernel at different scales for a pdf on the real line.
• Figure 5: Merge trees for modes: each pair shows the distribution on the circle from Figure \ref{['fig:BMT_example']} with modes indicated by $\star$. The BMTs are calculated with respect to diffusion distances depending on an exponential kernel that has a scale parameter $t>0$. The results shown are for various values of this parameter.

Theorems & Definitions (54)

• Example 1.1: Instability of Means
• Example 1.2: Instability of Medians
• Example 1.3: Stability of Barycentric Merge Trees
• Example 1.4: Modes and BMTs
• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Proposition 2.4
• proof