Persistent Diagram Estimation of Multivariate Piecewise Hölder-continuous Signals
Résumé
To our knowledge, the analysis of convergence rates for persistent diagram estimation from
noisy signals had remained limited to 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 persistent diagram (for sublevel sets filtration).
We show that for piecewise Hölder-continuous functions, with control over the reach of the
discontinuities set, taking the persistent diagram coming from a simple histogram estimator of
the signal, permit to achieve the minimax rates known for Hölder-continuous functions.
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