Wasserstein Distance-Preserving Vector Space of Persistent Homology

Analysis of large and dense networks based on topology is very difficult due to the computational challenges of extracting meaningful topological features from networks. In this paper, we present a computationally tractable approach to topological data analysis of large and dense networks. The approach utilizes principled theory from persistent homology and optimal transport to define a novel vector space representation for topological features. The feature vectors are based on persistence diagrams of connected components and cycles and are computed very efficiently. The associated vector space preserves the Wasserstein distance between persistence diagrams and fully leverages the Wasserstein stability properties. This vector space representation enables the application of a rich collection of vector-based models from statistics and machine learning to topological analyses. The effectiveness of the proposed representation is demonstrated using support vector machines to classify measured functional brain networks. Code for the topological vector space is available at https://github.com/topolearn.

Duke Scholars

Author Tananun Songdechakraiwut Computer Science