# Multiparameter Persistence and Nonlinear Hierarchical Clustering

Abstract: I will define a class of metrics on multiparameter persistence modules facilitated by the introduction of persistence contours. Using these, we can compute the feature counting invariant, which could previously not be computed, and show that it’s in general NP-hard to compute. Moreover, they can be used to put persistence modules in a machine learning context by providing a class of metrics that can be optimized over. This could for instance be used to solve classification problems in a metric learning sense, called contour learning. We then learn a one-parameter curve through the persistence module, yielding a nonlinear hierarchical clustering.