Topological Optimal Transport for Matching Persistent Homology Cycles Across Point Clouds

Core Concepts

The core message of this paper is to develop a mathematical framework for optimal transport-based matchings of topological features, by combining topological data analysis and optimal transport theory. The proposed Topological Optimal Transport (TpOT) framework provides a transport model on point clouds that minimises topological distortion while simultaneously yielding a geometrically informed matching between persistent homology cycles.

Abstract

The paper introduces the concept of a measure topological network, which integrates both geometric and topological information about a system, and defines a family of distances dTpOT,p between these objects. The key contributions are: Developing a flexible measure-theoretic formalism for simultaneously encoding the geometry and topology of a finite point cloud as a measure topological network. Showing that the distance dTpOT,p is a pseudometric on the space of measure topological networks, and characterizing the zero distance equivalence relation. Proving that the metric induced by dTpOT,p is geodesic, and showing that this metric space is non-negatively curved in the sense of Alexandrov. Providing efficient numerical algorithms for the TpOT optimization problem and demonstrating the framework on various examples. The TpOT framework addresses two key limitations of previous approaches: (1) it incorporates the significance (persistence) of topological features, and (2) it couples geometric and topological information via the PH-hypergraph structure. This allows for a topology-driven matching of point clouds that is geometrically informed.

Stats

The paper does not contain any explicit numerical data or statistics. It focuses on developing a theoretical framework for topological optimal transport.

Quotes

"The output is (1) a matching between points that is topologically driven and geometrically informed, coupled with (2) a matching between persistent homology classes that is geometrically driven and topologically informed." "Our framework is centred on an optimisation problem for which we provide efficient numerical algorithms, which we demonstrate on a variety of examples."

Key Insights Distilled From

Topological Optimal Transport for Geometric Cycle Matching

by Stephen Y Zh... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2403.19097.pdf

Deeper Inquiries

How can the TpOT framework be extended to handle time-varying or dynamic point cloud data

To extend the TpOT framework to handle time-varying or dynamic point cloud data, we can incorporate the concept of persistence over time. This can be achieved by considering a sequence of point clouds at different time steps and constructing a filtration that captures the evolution of topological features over time. By computing persistent homology on these time-varying point clouds, we can generate a series of persistence diagrams that represent the changing topological signatures. The TpOT framework can then be adapted to compare these dynamic topological networks by considering the evolution of geometric and topological structures over time. This extension would enable the analysis of how topological features evolve and interact in dynamic systems, providing valuable insights into temporal patterns and changes in complex data.

What are the potential applications of the TpOT framework beyond the examples provided in the paper

The TpOT framework has a wide range of potential applications beyond the examples provided in the paper. Some of the key applications include: Biomedical Imaging: TpOT can be used to analyze medical imaging data to identify topological features that are indicative of disease progression or treatment response. Robotics and Sensor Networks: TpOT can help in matching and comparing sensor data from different robots or IoT devices to understand spatial and temporal relationships in dynamic environments. Materials Science: TpOT can be applied to study the structural evolution of materials over time, aiding in the design of new materials with specific properties. Climate Science: TpOT can analyze spatiotemporal climate data to identify patterns and trends in weather systems and climate change. Financial Data Analysis: TpOT can be used to analyze time-varying financial data to detect anomalies, trends, and correlations in market behavior. These applications demonstrate the versatility and potential impact of the TpOT framework in various domains where understanding complex data structures is crucial.

How can the TpOT framework be integrated with other topological data analysis techniques, such as mapper or Reeb graphs, to provide a more comprehensive analysis of complex data

Integrating the TpOT framework with other topological data analysis techniques, such as mapper or Reeb graphs, can enhance the comprehensive analysis of complex data by combining different aspects of geometric and topological information. Here are some ways to integrate TpOT with other techniques: Mapper: Mapper is a method for topological data analysis that creates a graph-based representation of high-dimensional data. By incorporating TpOT into the Mapper algorithm, we can enhance the clustering and visualization of data points based on both geometric and topological similarities. This integration can provide a more holistic view of the data structure. Reeb Graphs: Reeb graphs capture the topological features of scalar fields by representing the evolution of level sets. By combining TpOT with Reeb graphs, we can analyze how the persistence of topological features changes over different levels of the scalar field. This integration can offer insights into the critical points and connectivity of the data. Persistent Homology: TpOT can be used in conjunction with persistent homology to analyze the evolution of topological features while minimizing distortion. By incorporating TpOT into the persistent homology pipeline, we can achieve a more robust and accurate analysis of complex datasets with both geometric and topological considerations. By integrating TpOT with other topological data analysis techniques, researchers can leverage the strengths of each method to gain a deeper understanding of complex data structures and relationships.