Topological Social Choice: Designing a Noise-Robust Polar Distance for Persistence Diagrams

TL;DR

This paper introduces a new topological distance metric based on polar coordinates for persistence diagrams, enhancing robustness and differentiability in noisy preference data analysis within social choice theory.

Contribution

It proposes a novel polar coordinate-based distance for persistence diagrams, addressing instability issues of classical metrics and enabling gradient-based learning in social choice applications.

Findings

Improved robustness of topological summaries under noise.

Enhanced stability and continuity of the new metric.

Successful application in supervised learning tasks.

Abstract

Topological Data Analysis (TDA) has emerged as a powerful framework for extracting robust and interpretable features from noisy high-dimensional data. In the context of Social Choice Theory, where preference profiles and collective decisions are geometrically rich yet sensitive to perturbations, TDA remains largely unexplored. This work introduces a novel conceptual bridge between these domains by proposing a new metric framework for persistence diagrams tailored to noisy preference data.We define a polar coordinate-based distance that captures both the magnitude and orientation of topological features in a smooth and differentiable manner. Our metric addresses key limitations of classical distances, such as bottleneck and Wasserstein, including instability under perturbation, lack of continuity, and incompatibility with gradient-based learning. The resulting formulation offers improved behavior in both theoretical and applied settings.To the best of our knowledge, this is the first study to systematically apply persistent homology to social choice systems, providing a mathematically grounded method for comparing topological summaries of voting structures and preference dynamics. We demonstrate the superiority of our approach through extensive experiments, including robustness tests and supervised learning tasks, and we propose a modular pipeline for building predictive models from online preference data. This work contributes a conceptually novel and computationally effective tool to the emerging interface of topology and decision theory, opening new directions in interpretable machine learning for political and economic systems.