Persistence Images: A Stable Vector Representation of Persistent Homology (1507.06217v3)

Abstract: Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.

• The paper establishes that transforming persistence diagrams into persistence images ensures stability under small perturbations.
• The paper demonstrates that persistence images achieve superior computational efficiency and classification performance compared to traditional methods like persistence landscapes.
• The paper validates the robustness and practical utility of persistence images through extensive experiments on diverse datasets in machine learning tasks.

Persistence Images: A Stable Vector Representation of Persistent Homology

Authors: Henry Adams, Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, Lori Ziegelmeier

Abstract: The paper focuses on transforming persistence diagrams (PDs), used in topological data analysis (TDA), into stable vector representations called persistence images (PIs). This research provides a rigorous proof of the stability of PIs and demonstrates their effectiveness in ML tasks across various datasets, showing enhancements over traditional methods.

Overview

Persistent homology is a central tool in TDA, capturing multiscale topological features of data. These features are often represented as persistence diagrams (PDs), which have limited utility in ML due to their inherent instability and lack of structure suitable for vector-based techniques. This paper introduces persistence images (PIs), a finite-dimensional vector representation of PDs, which maintain stability under perturbations and integrate seamlessly into ML workflows.

Core Contributions

1. Stable Transformation: The paper proves that the transformation of PDs to PIs is stable under small perturbations, a feature crucial for robustness in applications.
2. Comparative Performance: PIs show superior performance compared to existing methods like persistence landscapes (PLs) in tasks such as clustering and classification. They offer computational efficiency and accuracy advantages.
3. Experimental Validation: Through a series of experiments, PIs show effectiveness in classifying point cloud data of various topological shapes and dynamically generated data from models like the linked twist map and the anisotropic Kuramoto-Sivashinsky (aKS) equation.

Implications for Machine Learning

• Feature Discrimination: PIs facilitate enhanced feature discrimination, allowing for better classification performance. The authors leverage linear support vector machines (SVMs) to identify critical features, demonstrating practical applicability in ML.
• Computational Efficiency: The reduced computational complexity of generating and utilizing PIs compared to working directly with PDs indicates significant time savings, enabling more scalable data analysis.
• Parameter Robustness: The classification efficacy using PIs shows robust insensitivity to parameter choices like resolution and distribution variance, implying that practitioners can apply PIs without extensive parameter tuning.

Future Directions

The paper opens avenues for integrating persistent homology more deeply into ML and data-driven disciplines. Future work could explore:

• Advanced ML Techniques: Extending the use of PIs into more complex ML models, including deep learning architectures, to further enhance their utility in capturing topological features.
• Domain-Specific Applications: Applying PIs to a broader spectrum of scientific domains beyond the initial experiments, to fully leverage their stability and performance benefits.
• Parameter Optimization: Further exploration of parameter settings for specific data types could refine the efficacy and efficiency of PIs across different scenarios.

Conclusion

This research positions persistence images as a compelling tool in the TDA toolkit, marrying topological insights with vector space methodologies, thereby facilitating broader adoption and integration in machine learning applications.