Geometric averages of partitioned datasets

TL;DR

This paper presents a novel method for registering and analyzing partitioned datasets using geometric and optimal transport techniques, enabling meaningful summaries and visualizations of complex data ensembles.

Contribution

It introduces a partition-aware registration method based on Wasserstein space and local Fréchet means, with theoretical foundations and practical applications.

Findings

Effective registration of partitioned datasets demonstrated on political redistricting plans.

Theoretical results include curvature bounds and characterization of local means.

Method enables visualization and analysis of complex data ensembles.

Abstract

We introduce a method for jointly registering ensembles of partitioned datasets in a way which is both geometrically coherent and partition-aware. Once such a registration has been defined, one can group partition blocks across datasets in order to extract summary statistics, generalizing the commonly used order statistics for scalar-valued data. By modeling a partitioned dataset as an unordered $k$-tuple of points in a Wasserstein space, we are able to draw from techniques in optimal transport. More generally, our method is developed using the formalism of local Fr\'{e}chet means in symmetric products of metric spaces. We establish basic theory in this general setting, including Alexandrov curvature bounds and a verifiable characterization of local means. Our method is demonstrated on ensembles of political redistricting plans to extract and visualize basic properties of the space of plans for a particular state, using North Carolina as our main example.