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