A Kernel for Multi-Parameter Persistent Homology

Area of Science:

• Computational topology
• Machine learning
• Data science

Background:

• Topological data analysis (TDA) and persistent homology offer tools for analyzing high-dimensional, noisy datasets.
• Existing kernels connect one-parameter persistent homology to machine learning for tasks like shape analysis.
• A gap exists in applying multi-parameter persistence to machine learning for complex datasets.

Purpose of the Study:

• To develop a novel kernel construction for multi-parameter persistence.
• To integrate multi-parameter persistence with machine learning techniques for advanced data analysis.
• To establish a theoretical foundation for using topological features in multivariate data analysis.

Main Methods:

• Constructed a multi-parameter persistence kernel by weighting a one-parameter kernel along straight lines.
• Proved the stability and efficient computability of the proposed kernel.
• Demonstrated theoretical connections between TDA and machine learning for multivariate data.

Main Results:

• A novel, stable, and efficiently computable kernel for multi-parameter persistence was developed.
• The kernel facilitates the integration of multi-parameter topological features into machine learning models.
• Established a theoretical link between TDA and machine learning for analyzing complex, multivariate data.

Conclusions:

• The new kernel extends the applicability of persistent homology to multi-parameter settings in machine learning.
• This work provides a robust framework for leveraging topological information in multivariate data analysis.
• The findings pave the way for improved shape analysis, recognition, and classification using TDA.