Persistence Diagram Estimation : Beyond Plug-in Approaches

TL;DR

This paper introduces a new estimator for persistence diagrams in Topological Data Analysis that overcomes limitations of traditional plug-in methods, achieving consistency and parametric rates for certain signals in noisy settings.

Contribution

It proposes an image persistence-based estimator that relaxes regularity assumptions and provides theoretical guarantees in Gaussian white noise models.

Findings

Estimator is consistent for large classes of signals.

Achieves parametric convergence rates.

Applicable to piecewise-constant signals in noisy environments.

Abstract

Persistent homology is a tool from Topological Data Analysis (TDA) used to summarize the topology underlying data. It can be conveniently represented through persistence diagrams. Observing a noisy signal, common strategies to infer its persistence diagram involve plug-in estimators, and convergence properties are then derived from sup-norm stability. This dependence on the sup-norm convergence of the preliminary estimator is restrictive, as it essentially imposes to consider regular classes of signals. Departing from these approaches, we design an estimator based on image persistence. In the context of the Gaussian white noise model, and for large classes of piecewise-constant signals, we prove that the proposed estimator is consistent and achieves parametric rates.