Adaptive Partitioning for Template Functions on Persistence Diagrams

As the field of Topological Data Analysis continues to show success in theory and in applications, there has been increasing interest in using tools from this field with methods for machine learning. Using persistent homology, specifically persistence diagrams, as inputs to machine learning techniques requires some mathematical creativity. The space of persistence diagrams does not have the desirable properties for machine learning, thus methods such as kernel methods and vectorization methods have been developed. One such featurization of persistence diagrams by Perea, Munch and Khasawneh uses continuous, compactly supported functions, referred to as "template functions," which results in a stable vector representation of the persistence diagram. In this paper, we provide a method of adaptively partitioning persistence diagrams to improve these featurizations based on localized information in the diagrams. Additionally, we provide a framework to adaptively select parameters required for the template functions in order to best utilize the partitioning method. We present results for application to example data sets comparing classification results between template function featurizations with and without partitioning, in addition to other methods from the literature.Comment: To appear in proceedings of IEEE ICMLA 201

Persistence Bag-of-Words for Topological Data Analysis

Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs). PDs exhibit, however, complex structure and are difficult to integrate in today's machine learning workflows. This paper introduces persistence bag-of-words: a novel and stable vectorized representation of PDs that enables the seamless integration with machine learning. Comprehensive experiments show that the new representation achieves state-of-the-art performance and beyond in much less time than alternative approaches.Comment: Accepted for the Twenty-Eight International Joint Conference on Artificial Intelligence (IJCAI-19). arXiv admin note: substantial text overlap with arXiv:1802.0485

Data Analysis Methods using Persistence Diagrams

In recent years, persistent homology techniques have been used to study data and dynamical systems. Using these techniques, information about the shape and geometry of the data and systems leads to important information regarding the periodicity, bistability, and chaos of the underlying systems. In this thesis, we study all aspects of the application of persistent homology to data analysis. In particular, we introduce a new distance on the space of persistence diagrams, and show that it is useful in detecting changes in geometry and topology, which is essential for the supervised learning problem. Moreover, we introduce a clustering framework directly on the space of persistence diagrams, leveraging the notion of Fréchet means. Finally, we engage persistent homology with stochastic filtering techniques. In doing so, we prove that there is a notion of stability between the topologies of the optimal particle filter path and the expected particle filter path, which demonstrates that this approach is well posed. In addition to these theoretical contributions, we provide benchmarks and simulations of the proposed techniques, demonstrating their usefulness to the field of data analysis