Technology Category
Machine Learning: Learning Preferences or RankingsKnowledge Representation and Reasoning: PreferencesGame Theory and Economic Paradigms: Social Choice / Voting
Topological Social Choice: Designing a Noise-Robust Polar Distance for Persistence Diagrams
🤖 AI Summary
This paper addresses the challenge of topological modeling for noisy preference data in social choice. We propose a novel polar-coordinate persistence diagram distance—the first systematic application of persistent homology to preference spaces—encoding topological features’ magnitude and orientation in polar coordinates. This design overcomes key limitations of conventional metrics (e.g., bottleneck and Wasserstein distances), including instability, discontinuity, and non-differentiability under perturbations. The resulting metric is smooth and differentiable, enabling seamless integration into gradient-based learning frameworks. Coupled with a modular supervised learning paradigm, it supports online modeling of preference data. Experiments demonstrate substantial improvements in noise robustness and classification accuracy, while yielding interpretable characterizations of voting structures and preference dynamics—thereby enhancing downstream predictive tasks.
Technology Category
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Develops a noise-robust polar distance metric for persistence diagrams
Addresses instability and discontinuity in classical topological distances
Applies persistent homology to social choice systems for voting analysis
Innovation
Methods, ideas, or system contributions that make the work stand out.
Polar coordinate-based distance for persistence diagrams
Differentiable metric for topological feature analysis
Modular pipeline for predictive preference modeling
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