Persistence Diagram Estimation of Multivariate Piecewise Hölder-continuous Signals

To our knowledge, the analysis of convergence rates for persistence diagram estimation fromnoisy signals had predominantly relied on lifting signal estimation results through sup norm (orother functional norm) stability theorems. We believe that moving forward from this approachcan lead to considerable gains. We illustrate it in the setting of Gaussian white noise model. Weexamine from a minimax perspective, the inference of persistence diagram (for sublevel sets filtration). We show that for piecewise Hölder-continuous functions, with control over the reach ofthe discontinuities set, taking the persistence diagram coming from a simple histogram estimatorof the signal, permit to achieve the minimax rates known for Hölder-continuous functions.