The Topological Stability Index: A Variance-Based Measure for Persistence Barcodes

2026-05-28

AI Summary

This study addresses the lack of effective metrics in topological data analysis for quantifying the absolute dispersion of persistence barcode lifetimes, particularly under multiscale features and random perturbations. The authors propose the Topological Stability Index (TSI), which introduces a variance-based perspective into persistence homology summaries for the first time, yielding a scalar measure that is sensitive to absolute scale yet invariant under lifetime translations. They further develop its normalized variant, cvTSI, and a complementary Topological Signal Index (TSigI). Theoretical analysis establishes a monotonic affine relationship between cvTSI and the second-order Rényi entropy, thereby bridging variance-based and entropy-based topological summaries. Experiments demonstrate that TSI is highly sensitive to random noise and variations in lifetime magnitude, while remaining robust to deterministic trends, effectively overcoming limitations of conventional entropy-based approaches.

Abstract

We introduce the \emph{Topological Stability Index} (TSI), a variance-based scalar measure for persistence barcodes that quantifies the dispersion of persistence lifetimes. Unlike persistent entropy, which depends only on normalized weights, the TSI captures absolute variability and is sensitive to heterogeneous feature scales. We establish fundamental properties of the TSI, including its scaling behavior, invariance under lifetime translation and explicit update formulas under insertion and deletion of bars. We also consider a complementary first-moment-type quantity, the Topological Signal Index (TSigI), which captures the typical scale of persistence lifetimes and provides additional interpretability alongside the TSI. We further introduce a normalized version, $cv\text{TSI}$, which is scale invariant and admits an explicit algebraic relation to the Rényi entropy of order two. In particular, $cv\text{TSI}$ is an affine function of the collision probability $\sum_i p_i^2$, and therefore a monotone reparametrization of the Rényi entropy, providing a direct link between variance-based and entropy-based summaries in topological data analysis. Numerical experiments on synthetic data and stochastic time series demonstrate that the TSI captures structural variability complementary to entropy: it is relatively insensitive to deterministic trends, while responding strongly to stochastic fluctuations and variations in persistence magnitude.

Problem

• persistence barcodes
• topological data analysis
• variance-based measure
• topological stability
• persistent entropy

Innovation

• Topological Stability Index
• persistence barcodes
• variance-based measure
• Rényi entropy
• topological data analysis