# MS-COOT: Comparing Morse-Smale Complexes with Co-Optimal Transport

arXiv: https://arxiv.org/abs/2606.08258 · HTML: https://mlsanity.com/paper/2606.08258/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 57 references that are in the map.
- First posted on arXiv 6 Jun 2026.

## Builds on (most cited first, 23 of 23)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Optimal Transport for structured data with application on graphs](https://mlsanity.com/paper/1805.09114/) (2018)
- [Sliced Gromov-Wasserstein](https://mlsanity.com/paper/1905.10124/) (2019)
- [CO-Optimal Transport](https://mlsanity.com/paper/2002.03731/) (2020)
- [The Topology ToolKit](https://mlsanity.com/paper/1805.09110/) (2018)
- [Wasserstein Distances, Geodesics and Barycenters of Merge Trees](https://mlsanity.com/paper/2107.07789/) (2021)
- [Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization](https://mlsanity.com/paper/2106.00157/) (2021)
- [Hypergraph Co-Optimal Transport: Metric and Categorical Properties](https://mlsanity.com/paper/2112.03904/) (2021)
- [Lifted Wasserstein Matcher for Fast and Robust Topology Tracking](https://mlsanity.com/paper/1808.05870/) (2018)
- [Edit Distance between Merge Trees](https://mlsanity.com/paper/2207.08511/) (2022)
- [Unbalanced CO-Optimal Transport](https://mlsanity.com/paper/2205.14923/) (2022)
- [Intrinsic Interleaving Distance for Merge Trees](https://mlsanity.com/paper/1908.00063/) (2019)
- [Geometry-Aware Merge Tree Comparisons for Time-Varying Data with Interleaving Distances](https://mlsanity.com/paper/2107.14373/) (2021)
- [Branch Decomposition-Independent Edit Distances for Merge Trees](https://mlsanity.com/paper/2204.02919/) (2022)
- [Flexible and Probabilistic Topology Tracking with Partial Optimal Transport](https://mlsanity.com/paper/2302.02895/) (2023)
- [A Stable and Theoretically Grounded Gromov-Wasserstein Distance for Reeb Graph Comparison using Persistence Images](https://mlsanity.com/paper/2507.01171/) (2025)
- [Efficient approximation of Earth Mover's Distance Based on Nearest Neighbor Search](https://mlsanity.com/paper/2401.07378/) (2024)
- [Tracking Low-Level Cloud Systems with Topology](https://mlsanity.com/paper/2505.10850/) (2025)
- [Rapid and Precise Topological Comparison with Merge Tree Neural Networks](https://mlsanity.com/paper/2404.05879/) (2024)
- [A Comparative Study of Different Edit Distance-Based Methods for Feature Tracking using Merge Trees on Time-Varying Scalar Fields](https://mlsanity.com/paper/2509.17974/) (2025)
- [Comparing Morse Complexes Using Optimal Transport: An Experimental Study](https://mlsanity.com/paper/2309.04681/) (2023)
- [Fast Comparative Analysis of Merge Trees Using Locality Sensitive Hashing](https://mlsanity.com/paper/2409.08519/) (2024)

## Abstract (arXiv)

Understanding and comparing structures in scalar fields is a central challenge in scientific visualization, with applications ranging from feature analysis to temporal and structural comparison. The Morse-Smale (MS) complex provides a natural representation by decomposing a scalar field into regions induced by gradient flow. However, existing approaches typically rely on graph-based representations, capturing relationships between critical points while discarding region-level structure. In this work, we represent the MS complex as a hypergraph, where critical points form nodes and regions define hyperedges. We introduce MS-COOT, a co-optimal transport distance that jointly computes correspondences between critical points and regions. This formulation enables explicit region-to-region matching within a distance-based framework, allowing identification of region-level events such as splitting and merging. We instantiate this framework with domain-specific components, including a hypernetwork function encoding critical point-region relationships, persistence-based probability measures that emphasize topologically significant features, and a sample cost term that incorporates critical point attributes. We evaluate MS-COOT on five datasets spanning 2D simulations, 3D surface meshes, and volumetric data. Our results show that MS-COOT captures region-level structural changes that are not reflected by graph-based distances, while achieving strong performance in downstream tasks such as classification and resolution discrimination.