Research Briefing
math.ST 2605.29839v1 worth_reading

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

Joris Kirchner, Ioannis Diamantis

Published 2026-05-28 12:21:36 相关性 0.7000 价值 0.6800 math.ST cs.IT physics.data-an stat.ML
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摘要

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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.

相关性判断

medium
相关方向
information_theory topological_data_analysis statistical_signal_processing
判断依据

Primarily a Topological Data Analysis paper, but it has a direct information-theoretic connection via an exact relation between the normalized TSI and Rényi entropy of order two. Relevant as an adjacent statistical/information-theoretic methods paper rather than core information theory or communications.

价值判断

Medium relevance: the paper is mainly topological data analysis, but the explicit affine relation between normalized TSI and collision probability/Rényi entropy gives it clear adjacent value for information-theoretic methods. Structure evidence suggests solid technical content: properties, update formulas, bounds, normalization, and experiments are identified with high confidence. Likely useful as a methods paper for tracking entropy-like summaries beyond standard persistent entropy, but not urgent enough for deep review as core cs.IT work.

核心问题与主要方法

核心问题

summarize persistence barcodes with a scalar that captures absolute dispersion of lifetimes, not just normalized relative distribution

场景：topological data analysis on persistence barcodes from geometric data and stochastic time series

主要方法

Define TSI as the unbiased sample variance of persistence lifetimes, so it captures absolute heterogeneity among bars rather than normalized distribution alone. Use elementary variance algebra to derive scaling, translation invariance, extremal configurations under fixed total persistence, and exact insertion/deletion updates. Introduce TSigI as a persistence-weighted mean lifetime, giving a complementary first-moment or signal-scale descriptor paired with TSI's second-order dispersion. Normalize TSI by the squared mean lifetime to obtain cvTSI, making the statistic scale invariant and expressible through normalized lifetime weights. Relate cvTSI algebraically to collision probability sum_i p_i^2 and hence to Rényi entropy of order two, positioning it as an entropy-equivalent concentration measure. Analyze perturbation behavior through Wasserstein/bottleneck bounds in restricted settings and through explicit short-bar insertion limits.

关键贡献与后续阅读

关键贡献

Introduces TSI as a variance-based scalar statistic for persistence barcodes that distinguishes equal-total-persistence barcodes with different lifetime dispersion. Establishes basic structural properties: quadratic scaling under uniform filtration scaling, invariance under uniform lifetime translation, and extremal behavior for fixed bar count and total persistence. Derives explicit update formulas for insertion and deletion of bars, including a deviation-from-mean threshold for whether a new bar increases TSI. Shows limits of stability by proving discontinuity under insertion of arbitrarily short bars, while still giving quantitative bounds in special Wasserstein or bottleneck comparison settings. Defines TSigI and a hierarchy of topological signal moments based on consecutive power sums, relating TSI to the gap between ordinary and persistence-weighted mean lifetimes. Defines normalized cvTSI and proves its exact algebraic relation to collision probability and Rényi entropy of order two. Provides numerical evidence on synthetic geometric data, noise models, Alpha/Rips complexes, and geometric Brownian motion that TSI captures structural variability complementary to persistent entropy.

研究启发

How much of cvTSI's information-theoretic content is new beyond recognizing that variance of normalized lifetimes is equivalent to collision probability? Do the numerical experiments include quantitative comparisons against downstream tasks such as classification, anomaly detection, or hypothesis testing, or are they mainly illustrative? How sensitive are conclusions to the choice of filtration and to preprocessing that removes short-lived bars? Can the proposed functional analogue of TSI within persistence curves recover stability while preserving the dispersion interpretation?

限制与不确定性

Information-theory contribution appears mostly a reparameterization link to Rényi entropy rather than a new information-theoretic result. Scalar barcode summaries may have limited downstream impact, and listed limitations include sensitivity to short-bar insertions and compression of filtration dynamics.

原文信息

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最近更新 2026-05-30 13:21
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Abstract

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

We introduce the 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, c ​ v ​ TSI cv\textup{TSI} , which is scale invariant and admits an explicit algebraic relation to the Rényi entropy of order two. In particular, c ​ v ​ TSI cv\textup{TSI} is an affine function of the collision probability ∑ i p i 2 \sum_{i}p_{i}^{2} , and therefore a monotone reparam
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