Selecting Interpretable Circular Coordinates from Data

2026-07-09

AI Summary

While existing circular coordinates derived from persistent cohomology can reveal cyclic structures in data, they often lack scientific interpretability and fail to correspond to physical quantities such as angles or phases. This work proposes a novel approach to select interpretable circular coordinates from a user-provided dictionary of candidate functions by formulating the problem—under a minimum-energy basis criterion—as a minimum-weight basis problem in a vector matroid in the continuous setting, and extends this framework to discrete point clouds. By incorporating domain knowledge into the selection of circular coordinates, our method bridges topological features with physical meaning. We introduce the CIRCOL algorithm and a consistent L² inner product estimator for cochains. Experiments on synthetic data, molecular dynamics trajectories, and neural recordings demonstrate that our approach accurately identifies interpretable coordinates and effectively diagnoses persistent cohomology classes that are either meaningless or not yet explained.

Abstract

Circular coordinates obtained from persistent cohomology reveal loop structure in data, but they usually remain abstract: A detected circle does not tell us which measured angle, phase, torsion, or decoder explains it. We propose a method for selecting interpretable circle-valued coordinates from a user-supplied dictionary of scientifically meaningful candidates explaining the detected cohomology. In the continuous setting, each candidate is represented by the cohomology class of its pulled-back angular form, and selecting a minimum-energy set of candidates spanning the relevant $H^1$ subspace becomes a minimum-weight basis problem in a vector matroid. We then introduce CIRCOL, a method for discrete point clouds sampled from the manifold. We prove that the introduced cochain inner product is a consistent estimator of the $L^2$ inner product of fixed smooth 1-forms under non-uniform sampling. The resulting projection matrix both helps selecting a basis of low-energy dictionary coordinates and diagnoses topologically trivial candidates or unexplained persistent classes. Finally, we verify the effectiveness of our method on synthetic examples, on molecular simulations, and neural recordings of head-direction cells.

Problem

• interpretable circular coordinates
• persistent cohomology
• circle-valued functions
• topological data analysis
• scientific interpretability

Innovation

• interpretable circular coordinates
• persistent cohomology
• vector matroid
• CIRCOL
• cochain inner product