Alessio Borgi

Alessio Borgi

PhD in Graph Neural Networks and Generative AI 👨🏻‍💻🤖

Circular Coordinates: Parameterising Periodic Structure in Data

3 minute read

Published:

TL;DR: Persistent H¹ detects loops in data. Circular coordinates (de Silva, Vejdemo-Johansson & Carlsson, 2011) upgrade this from a binary "loop exists" to a quantitative map f: P → S¹. A persistent 1-cocycle is smoothed (made harmonic) then integrated to a circle-valued function. Applications include gait parameterisation, gene expression cycles, and neural place-cell topology.

From Detection to Parameterisation

Standard TDA asks: “does the data have a loop?” But for a point cloud living near a circle — e.g., motion capture data in a gait cycle — we want more: a continuous function \(f: P \to S^1\) that assigns each data point a phase in the cycle.

This requires:

  1. Detecting the loop via persistent \(H_1\) (a long-lived bar).
  2. Extracting a representative 1-cocycle \(\varphi\) for that class.
  3. Making \(\varphi\) harmonic (smooth/consistent across the complex).
  4. Integrating \(\varphi\) to a map \(f: P \to S^1\).

Step 1: Persistent 1-Cocycle

Compute persistent cohomology \(H^1\) of the Rips filtration \(\mathrm{Rips}(P, r)\) at the scale \(r^*\) where the longest bar is born. The corresponding cocycle \(\varphi \in Z^1\) assigns a value in \(\mathbb{Z}\) (or \(\mathbb{R}/\mathbb{Z}\)) to each edge.

Over \(\mathbb{R}\): \(\varphi([p_i, p_j]) \in \mathbb{R}\) with \(\delta \varphi = 0\) (cocycle condition: \(\varphi([p_j,p_k]) - \varphi([p_i,p_k]) + \varphi([p_i,p_j]) = 0\) on all triangles).

Step 2: Harmonic Smoothing

A cocycle \(\varphi\) is not unique in its cohomology class (adding any coboundary \(\delta \psi\) gives an equivalent cocycle). Choose the harmonic representative: the unique cocycle in its cohomology class that minimises the \(\ell^2\) norm.

Practically: solve the Laplacian system \(\Delta \varphi = 0\) subject to \(\varphi\) representing the correct cohomology class. This gives a smooth, geometrically meaningful cocycle.

Step 3: Integration to Circle

With the harmonic cocycle, define the circular coordinate \(f: P \to S^1 = \mathbb{R}/\mathbb{Z}\) by:

$$f(p_j) - f(p_i) \equiv \varphi([p_i, p_j]) \pmod{1}$$

Fix \(f(p_0) = 0\) and propagate consistently around any spanning tree. The map is well-defined modulo 1 because \(\varphi\) is a cocycle.

Applications

  • Gait analysis: motion capture data of human walking lives near \(S^1\); circular coordinates give a clean phase parameter.
  • Gene expression cycles: circadian or cell-cycle data. Singh et al. (2008) found \(S^1\) structure in human primary visual cortex data.
  • Neural data: place cells in the hippocampus encode spatial position; their joint firing patterns have the topology of a torus \(T^2 = S^1 \times S^1\). Circular coordinates on each \(S^1\) factor give spatial coordinates.
Key Insight: Circular coordinates are a completely data-driven parameterisation of periodicity — they require no prior knowledge of the cycle length, no phase alignment, and no embedding. The TDA-derived map is guaranteed to be consistent (a well-defined map to S¹) whenever the underlying H¹ class is non-trivial. This is strictly stronger than what PCA or manifold learning methods can provide.

References

  • V. de Silva, D. Vejdemo-Johansson, G. Carlsson, “Persistent Cohomology and Circular Coordinates,” Discrete & Computational Geometry, 2011. arXiv:0905.4887.
  • G. Singh, F. Mémoli, G. Carlsson, “Topological Methods for the Analysis of High Dimensional Data Sets,” SPBG, 2007.

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