Metrics and Stabilization in One Parameter Persistence

We propose the use of persistent homology in a supervised way. We believe homological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics. Choosing a pseudometric between persistent vector spaces leads to a model. Fitting this model is what we believe supervised homological persistence is. We develop theory behind constructing such models and we give evidence of the usefulness of this approach in concrete data analysis tasks.