Topological Data Analysis with Combinatorial Laplacian for Data Clustering

Abstract

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This thesis attempts to combine machine learning and topological data analysis (TDA). We exam the machine that only learned the original data without interruption to face various testing data under linear transformation by adding Betti number as an additional feature. Our experiments are based on the theory of homology group by constructing simplicial complexes of images and the discrete version of the Hodge theorem with higher-order Laplacian matrices. This approach performs well and representsthe importance concerning topological structure of the image itself. We believe that TDA is a good supporter to help machine learning models dealing with more complicated data rather than pouring more and more different cases for training. In the future, we would pay more attention to the application and the theory of TDA combined with diverse models.