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Spherical Coordinates from Persistent Cohomology

Nikolas C. Schonsheck, Stefan C. Schonsheck

TL;DR

The paper develops spherical parameterizations for data by leveraging second-degree persistent cohomology to produce a sphere-valued map $X o S^2$. It constructs an initial lift on the $3$-skeleton from a $2$-cocycle, then refines it via an energy-based smoothing framework that includes discrete harmonic and spring energies, with an alternating gradient/Möbius update that preserves the underlying topology. It proves that the smoothing preserves the relevant homotopy class and demonstrates the approach on synthetic and real datasets, including multi-sphere and object-rotation scenarios, yielding topology-faithful, visualization-friendly embeddings. The method offers a topologically grounded NLDR tool that yields exact spherical embeddings in the presence of noise and irregular sampling, with practical implications for visualization and downstream analysis.

Abstract

We describe a method to obtain spherical parameterizations of arbitrary data through the use of persistent cohomology and variational optimization. We begin by computing the second-degree persistent cohomology of the filtered Vietoris-Rips (VR) complex of a data set $X$ and extract a cocycle $α$ from any significant feature. From this cocycle, we define an associated map $α: VR(X) \to S^2$ and use this map as an infeasible initialization for a variational model, which we show has a unique solution (up to rigid motion). We then employ an alternating gradient descent/Möbius transformation update method to solve the problem and generate a more suitable, i.e., smoother, representative of the homotopy class of $α$, preserving the relevant topological feature. Finally, we conduct numerical experiments on both synthetic and real-world data sets to show the efficacy of our proposed approach.

Spherical Coordinates from Persistent Cohomology

TL;DR

The paper develops spherical parameterizations for data by leveraging second-degree persistent cohomology to produce a sphere-valued map . It constructs an initial lift on the -skeleton from a -cocycle, then refines it via an energy-based smoothing framework that includes discrete harmonic and spring energies, with an alternating gradient/Möbius update that preserves the underlying topology. It proves that the smoothing preserves the relevant homotopy class and demonstrates the approach on synthetic and real datasets, including multi-sphere and object-rotation scenarios, yielding topology-faithful, visualization-friendly embeddings. The method offers a topologically grounded NLDR tool that yields exact spherical embeddings in the presence of noise and irregular sampling, with practical implications for visualization and downstream analysis.

Abstract

We describe a method to obtain spherical parameterizations of arbitrary data through the use of persistent cohomology and variational optimization. We begin by computing the second-degree persistent cohomology of the filtered Vietoris-Rips (VR) complex of a data set and extract a cocycle from any significant feature. From this cocycle, we define an associated map and use this map as an infeasible initialization for a variational model, which we show has a unique solution (up to rigid motion). We then employ an alternating gradient descent/Möbius transformation update method to solve the problem and generate a more suitable, i.e., smoother, representative of the homotopy class of , preserving the relevant topological feature. Finally, we conduct numerical experiments on both synthetic and real-world data sets to show the efficacy of our proposed approach.
Paper Structure (20 sections, 6 theorems, 22 equations, 11 figures, 1 algorithm)

This paper contains 20 sections, 6 theorems, 22 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Main Pipeline
  4. Acknowledgments
  5. Persistent Cohomology
  6. Background on Persistent Cohomology
  7. Lifting to Integer Coefficients
  8. Obtaining a Map to the 2-Sphere
  9. Smoothing the Initial Map
  10. Harmonic Energy
  11. Spring Energy
  12. Optimization Routine
  13. Relation to the 1D Problem
  14. Energy Minimization Preserves Homotopical Information
  15. Numerical Results
  16. ...and 5 more sections

Key Result

Proposition 6

The map $\hat{\alpha}$ defined on $\mathop{\mathrm{sk}}\nolimits_2(X)$ in Definition defn_canonical_lift can be extended to $\mathop{\mathrm{sk}}\nolimits_3(X)$.

Figures (11)

  • Figure 1: Nonlinear dimensionality reduction of a noisy sphere embedded in $\mathbb{R}^{50}$ using several popular techniques and our methodology.
  • Figure 2: Given a 2-cocycle $\alpha$, we consider the corresponding map on the boundary of a generic 3-simplex $\sigma$.
  • Figure 3: Left: Our choice of canonical directions for positive and negative orientations for a triangle with nonzero winding ($w = \pm 1)$, vertices at $[1,0,0]$ and barycenter $[-1,0,0]$. The outward normal is chosen to be positive and the inward negative. Right: Equilateral Triangle mapped to circle and regular, triangular pyramid mapped to the unit sphere using harmonic energy minimizing maps
  • Figure 4: First Row: Evenly sampled circle, Second Row: Noisy, sparsely sampled circle, Third Row: Trefoil knot in 3D (visualized in 2D) with spring energy, Fourth row: Noisy Ellipse in 50D (visualized in 2D without noise) with harmonic energy, Last row: Same as above but with spring energy
  • Figure 5: First Row: Sphere Second Row: Sphere with noise, Third Row: Ellipse
  • ...and 6 more figures

Theorems & Definitions (19)

  • Definition 1
  • Remark 2
  • Definition 3
  • Remark 4
  • Definition 5
  • Proposition 6
  • proof
  • Definition 7
  • Definition 8
  • Proposition 9
  • ...and 9 more