We prove that persistence diagrams with the p -Wasserstein distance is the universal p -subadditive commutative monoid on an underlying metric space with a distinguished subset. This result applies to persistence diagrams, to barcodes, and to multiparameter persistence modules. In addition, the 1-Wasserstein distance satisfies Kantorovich-Rubinstein duality.

We use topological data analysis (TDA) to study how data transforms as it passes through successive layers of a deep neural network (DNN). We compute the persistent homology of the activation data for each layer of the network and summarize this information using persistence landscapes. The resulting feature map provides both an informative visualization of the network and a kernel for statistical analysis and machine learning. A statistical test shows that it correlates with classification accuracy. We observe that the topological complexity often increases with training and that the topological complexity does not decrease with each layer.