Persistence Diagrams Estimation of Multivariate Piecewise H{\"o}lder-continuous Signals

2025-12-31 · Source: JMLR · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Expert, quick

Summary

Hugo Henneuse's 2026 paper, "Persistence Diagrams Estimation of Multivariate Piecewise H{"o}lder-continuous Signals," introduces a novel approach for estimating persistence diagrams from noisy signals, particularly in nonparametric regression settings. The work challenges the traditional reliance on sup-norm stability theorems for convergence rate analysis, proposing algebraic stability instead. This method directly targets the bottleneck distance via interleaving, enabling the incorporation of deformation retractions of sublevel sets. For piecewise H{"o}lder-continuous functions with controlled discontinuity reach, using a simple histogram estimator for the signal allows achieving minimax rates previously known only for H{"o}lder-continuous functions. This advancement addresses boundary discontinuities that sup-norm stability analyses cannot handle.

Key takeaway

For research scientists working on topological data analysis or nonparametric regression with noisy, piecewise H{"o}lder-continuous signals, you should consider adopting algebraic stability methods over traditional sup-norm approaches. This shift can lead to more accurate persistence diagram estimations, especially when dealing with boundary discontinuities, by achieving minimax rates with simpler estimators.

Key insights

Algebraic stability offers considerable gains over sup-norm stability for persistence diagram estimation from noisy signals.

Principles

• Algebraic stability improves bottleneck distance targeting.
• Deformation retractions accommodate boundary discontinuities.

Method

The method involves using a simple histogram estimator for piecewise H{"o}lder-continuous functions, leveraging algebraic stability and deformation retractions to achieve minimax rates for persistence diagram estimation.

In practice

• Apply histogram estimators for piecewise H{"o}lder signals.
• Utilize algebraic stability for improved diagram inference.

Topics

• Persistence Diagrams
• Convergence Rates
• Piecewise H{"o}lder-continuous Functions
• Algebraic Stability
• Bottleneck Distance

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by JMLR.