Topological Machine Learning with Persistence Indicator Functions

Abstract

Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional scaling, while providing a firm theoretical ground. Many modern machine learning algorithms, however, are based on kernels. This paper presents persistence indicator functions (PIFs), which summarize persistence diagrams, i.e., feature descriptors in topological data analysis. PIFs can be calculated and compared in linear time and have many beneficial properties, such as the availability of a kernel-based similarity measure. We demonstrate their usage in common data analysis scenarios, such as confidence set estimation and classification of complex structured data.

arXiv commentTopology-based Methods in Visualization 2017

References in corpus (3)

• Scikit-learn: Machine Learning in Python
• Statistical topological data analysis using persistence landscapes
• Geometry Helps to Compare Persistence Diagrams

Cited by in corpus (8)

• On the stability of persistent entropy and new summary functions for TDA
• Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization
• Progressive Wasserstein Barycenters of Persistence Diagrams
• Fuzzy c-Means Clustering for Persistence Diagrams
• A computationally efficient framework for vector representation of persistence diagrams
• The magnitude vector of images
• Characterizing and Measuring the Similarity of Neural Networks with Persistent Homology
• A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering