Parametrization of Symmetry in Data

TL;DR

This paper introduces a comprehensive framework for analyzing and quantifying symmetries in data configurations over parameters, utilizing category theory, invariants like symmetry barcodes, and algorithms for practical computation.

Contribution

It develops new invariants and theoretical tools for persistent symmetry analysis, bridging geometric group theory, topological data analysis, and machine learning.

Findings

Defined persistent symmetry groups and invariants

Established stability theorems for symmetry barcodes

Provided algorithms for symmetry analysis in low-dimensional data

Abstract

Symmetry plays a fundamental role in understanding natural phenomena and mathematical structures. This work develops a comprehensive theory for studying the persistent symmetries and degree of asymmetry of finite point configurations over parameterization in metric spaces. Leveraging category theory and span categories, we define persistent symmetry groups and introduce novel invariants called symmetry barcodes and polybarcodes that capture the birth, death, persistence, and reappearance of symmetries over parameter evolution. Metrics and stability theorems are established for these invariants. The concept of symmetry types is formalized via the action of isometry groups in configuration spaces. To quantitatively characterize symmetry and asymmetricity, measures such as degree of symmetry and symmetry defect are introduced, the latter revealing connections to approximate group theory in Euclidean settings. Moreover, a theory of persistence representations of persistence groups is developed, generalizing the classical decomposition theorem of persistence modules. Persistent Fourier analysis on persistence groups is further proposed to characterize dynamic phenomena including symmetry breaking and phase transitions. Algorithms for computing symmetry groups, barcodes, and symmetry defect in low-dimensional spaces are presented, complemented by discussions on extending symmetry analysis beyond geometric contexts. This work thus bridges geometric group theory, topological data analysis, representation theory, and machine learning, providing novel tools for the analysis of the parametrized symmetry of data.