Differentiable and Measure-Based Mapper

In this talk, I will discuss two recent studies on Mapper graphs. In the first one, we build on a recently proposed optimization framework incorporating topology to provide the first filter optimization scheme for Mapper graphs. In order to achieve this, we propose a relaxed and more general version of the Mapper graph, whose convergence properties are investigated. In the second one, we focus on finding an appropriate, density-aware, metric for comparing Reeb and Mapper graphs seen as metric measure spaces, in order to, e.g., quantify the rate of convergence of the Mapper graph to the Reeb graph. We focus on the use of Gromov-Wasserstein metrics to compare these graphs directly in order to better incorporate the probability measures that data points are sampled from.