[Paper Review] Persistence Flamelets: multiscale Persistent Homology for kernel density exploration

Tullia Padellini, Pierpaolo Brutti|arXiv (Cornell University)|Sep 20, 2017

Topological and Geometric Data Analysis18 references3 citations

TL;DR

This paper introduces Persistence Flamelets, a multiscale topological summary derived from persistent homology, to analyze kernel density estimators (KDEs) across bandwidths. By tracking persistent topological features (e.g., loops) across bandwidth scales, it enables topologically informed bandwidth selection, significantly improving detection of low-dimensional structures like plate boundaries in seismic data compared to standard methods.

ABSTRACT

In recent years there has been noticeable interest in the study of the "shape of data". Among the many ways a "shape" could be defined, topology is the most general one, as it describes an object in terms of its connectivity structure: connected components (topological features of dimension 0), cycles (features of dimension 1) and so on. There is a growing number of techniques, generally denoted as Topological Data Analysis, aimed at estimating topological invariants of a fixed object; when we allow this object to change, however, little has been done to investigate the evolution in its topology. In this work we define the Persistence Flamelets, a multiscale version of one of the most popular tool in TDA, the Persistence Landscape. We examine its theoretical properties and we show how it could be used to gain insights on KDEs bandwidth parameter.

Motivation & Objective

• To develop a multiscale topological summary that captures how topological features in kernel density estimators evolve with bandwidth.
• To address the limitation of standard bandwidth selection methods in detecting low-dimensional structures such as loops or ridges in data.
• To provide a principled, topologically informed heuristic for bandwidth selection that emphasizes persistent, meaningful features in the density.
• To extend topological data analysis (TDA) beyond static point clouds to families of parametric objects, such as KDEs across bandwidths.
• To enable better visualization and interpretation of multidimensional time series and complex density structures through topological persistence.

Proposed method

• Persistence Flamelets are constructed as a multiscale extension of the persistence landscape, summarizing the persistence of topological features (e.g., loops) across a range of bandwidths.
• For each bandwidth h, the KDE is computed, and its superlevel sets are analyzed via persistent homology to extract Betti numbers and persistence diagrams.
• The method tracks the birth and death of topological features (e.g., loops) across bandwidths, encoding their persistence as a function of scale.
• The resulting flamelet representation aggregates persistence values across bandwidths, forming a continuous, interpretable summary of topological significance.
• Bandwidth selection is guided by maximizing the Persistence Flamelet value, favoring bandwidths that preserve the most persistent features.
• The approach is validated on seismic data, comparing topologically aware bandwidths against Silverman’s rule of thumb and plug-in estimators.

Experimental results

Research questions

• RQ1 How can topological features in kernel density estimators be tracked across varying bandwidths to inform bandwidth selection?
• RQ2 Can persistence-based summaries improve the detection of low-dimensional structures such as loops or ridges in complex, high-dimensional data?
• RQ3 How does the choice of bandwidth affect the topological structure of KDEs, and can this be quantified via a multiscale topological summary?
• RQ4 To what extent do topologically informed bandwidths outperform classical methods in recovering known structural features in real-world data?
• RQ5 Can Persistence Flamelets serve as a reliable, probabilistically grounded tool for statistical inference beyond exploratory data analysis?

Key findings

• The topologically aware bandwidth $\widehat{h}_{\text{TA}}$, selected by maximizing the Persistence Flamelet, successfully highlights the Philippine plate boundary in seismic data, where over 26% of seismic activity was concentrated.
• In contrast, Silverman’s Normal Rule and plug-in bandwidths failed to recover any plate boundaries, indicating their insensitivity to low-dimensional topological features.
• The Persistence Flamelets revealed one dominant loop feature with significantly higher persistence than others, indicating a structurally prominent feature in the data.
• The method demonstrated that features persisting across multiple bandwidth scales are more likely to represent meaningful, stable structures in the underlying density.
• The flamelet-based bandwidth selection outperformed standard heuristics in capturing the true topological structure of the data, especially in cases with singular or low-dimensional support.
• The framework provides a probabilistically grounded, interpretable alternative to cross-validation in settings where cross-validation fails due to singular densities.