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A Kernel for Multi-Parameter Persistent Homology.

René Corbet¹, Ulderico Fugacci¹, Michael Kerber¹

¹Graz University of Technology, Austria.

Computers & Graphics: X

|December 28, 2020

Summary

This summary is machine-generated.

Topological data analysis uses persistent homology to analyze complex data. This study introduces a new kernel for multi-parameter persistence, enhancing machine learning for multivariate data analysis and shape recognition.

Keywords:

Machine Learning Multivariate Analysis Persistent Homology Topological Data Analysis

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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.