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arxiv: 2605.29839 · v1 · pith:HL7UIJKBnew · submitted 2026-05-28 · 🧮 math.ST · cs.IT· math.IT· physics.data-an· stat.ML· stat.TH

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

Joris Kirchner , Ioannis Diamantis This is my paper

Pith reviewed 2026-06-29 00:24 UTC · model grok-4.3

classification 🧮 math.ST cs.ITmath.ITphysics.data-anstat.MLstat.TH
keywords Topological Stability IndexPersistence BarcodesRényi EntropyCollision ProbabilityVariance MeasureTopological Data AnalysisNormalized IndexLifetime Dispersion
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The pith

The normalized Topological Stability Index on persistence barcodes is an affine function of the collision probability and thus a monotone reparametrization of Rényi entropy of order two.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the Topological Stability Index as a variance measure on the lifetimes in a persistence barcode, capturing absolute dispersion rather than only relative proportions. A scale-invariant version called cvTSI is shown to equal an affine transform of the sum of squared normalized lifetimes. This identity directly connects the variance-based index to the Rényi entropy of order two. The authors also introduce a first-moment companion called TSigI and derive scaling, translation-invariance, and update rules for the measures. Experiments indicate that the index responds to magnitude variations and stochastic fluctuations where entropy summaries stay flat.

Core claim

The normalized cvTSI is an affine function of the collision probability sum p_i squared, and therefore a monotone reparametrization of the Rényi entropy of order two, establishing an explicit algebraic bridge between variance-based and entropy-based summaries for persistence barcodes.

What carries the argument

cvTSI, the normalized coefficient of variation of persistence lifetimes, which satisfies an algebraic identity with the collision probability sum p_i squared.

If this is right

  • TSI remains sensitive to heterogeneous feature scales while persistent entropy does not.
  • Explicit update formulas allow efficient recomputation when bars are inserted or deleted.
  • TSigI supplies the typical lifetime scale as a complement to the dispersion measured by TSI.
  • The index detects stochastic fluctuations in time series that leave entropy unchanged.
  • Translation invariance under lifetime shifts follows directly from the variance definition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The algebraic link suggests that existing Rényi-entropy algorithms could be reused to compute variance summaries on barcodes without separate code paths.
  • Hybrid descriptors that combine cvTSI with higher-order Rényi entropies might separate scale effects from distribution shape more cleanly than either family alone.
  • Because the relation is exact, any theoretical guarantee proved for one summary transfers immediately to the other after the affine reparametrization.

Load-bearing premise

Normalizing the lifetimes to form a probability distribution does not erase the absolute-scale information that the original variance measure is meant to retain.

What would settle it

A set of bar lengths for which the computed cvTSI differs numerically from one minus the sum of squared normalized lengths.

Figures

Figures reproduced from arXiv: 2605.29839 by Ioannis Diamantis, Joris Kirchner.

Figure 1
Figure 1. Figure 1: Two persistence barcodes with identical total persistence but different distri￾butions of lifetimes. The top barcode exhibits moderate variability, with one long bar and several shorter ones, whereas the bottom barcode shows a highly uneven distribution, concentrating persistence in a small number of longer bars while the remaining bars are significantly shorter. This highlights that the TSI measures dispe… view at source ↗
Figure 2
Figure 2. Figure 2: Extremal configurations for the TSI under fixed total persistence L and number of bars n. A barcode with equal bar lengths ℓi = L/n minimizes the TSI (left), while a barcode in which one bar has length L and the remaining n−1 bars have length zero (shown schematically) maximizes the TSI (right). 2.2. Behaviour under Scaling and Translation. We now examine how TSI behaves under simple transformations of the… view at source ↗
Figure 3
Figure 3. Figure 3: Effect of uniform scaling and death shifts on a barcode. Uniform scaling mul￾tiplies all lifetimes by the same factor and causes the TSI to scale quadratically, whereas shifting all death times by a constant leaves the TSI unchanged. 2.3. Addition and Removal of Bars. We now describe how the TSI changes under insertion or deletion of a bar. To do this, we need the following lemma. Lemma 2.2. Let B be a bar… view at source ↗
Figure 4
Figure 4. Figure 4: Effect of inserting a new bar. If the new bar length is close to the mean lifetime, the TSI may decrease; if it is sufficiently far from the mean, the TSI increases. This threshold behavior is quantified by the variance barrier in the preceding corollary. This sensitivity to the position of the inserted bar has an important consequence for stability. As a consequence, if ℓ → 0, TSI(B ∪ {[b, b + ℓ)}) → n B … view at source ↗
Figure 5
Figure 5. Figure 5: Non-continuity of the TSI under insertion of arbitrarily short bars. As the added bar length ℓ tends to zero, the Wasserstein distance to the original barcode vanishes, while the TSI converges to a value that in general differs from the original one. Lemma 2.4 (Bounds relative to the empty diagram). Let B = {[bi , di)} n B i=1 be a barcode. Then, for every 2 ≤ p ≤ ∞, one has (10) TSI(B) ≤ 4(n B) −2/p dWp (… view at source ↗
Figure 6
Figure 6. Figure 6: Two barcodes with the same normalized lifetime distribution (1/3, 2/3) and therefore the same persistent entropy, but different total scale. The right barcode is obtained by uniformly scaling the lifetimes of the left by a factor of two. Persistent entropy is invariant under such scaling, whereas the TSI scales quadratically. since x log x → 0 as x → 0 +. Thus, persistent entropy provides a scale-invariant… view at source ↗
Figure 7
Figure 7. Figure 7: Extremal behavior of the normalized TSI and R´enyi entropy of order two. Uniform lifetime distributions minimize cvTSI and maximize H2, whereas maximally con￾centrated lifetime distributions maximize cvTSI and minimize H2. Proposition 3.5. Let p = (p1, . . . , pnB ) be a probability vector of the form pi = 1 nB + εi , X i εi = 0, with ∥ε∥ sufficiently small. Then E(B) = log(n B) − n B − 1 2nB cvTSI(B) + O(… view at source ↗
Figure 8
Figure 8. Figure 8: Two disjoint circles with radii 1 and 1 4 . (A) Point cloud sampled with increasing density. (B) Persistence diagram showing two dominant H1 features corresponding to the two loops. (C) Evolution of TSI, persistent entropy, and cvTSI as a function of sample size. Intertwined circles: We next consider two intersecting circles, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Two intertwined circles of equal radius. (A) Point cloud. (B) Persistence diagram showing multiple H1 features arising from the overlapping structure. (C) Evolution of TSI, persistent entropy, and cvTSI as a function of sample size [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Random sampling on intertwined circles. (A) A random realization of the sampled point cloud. (B) Corresponding persistence diagram, illustrating variability due to stochastic sampling. (C) Monte Carlo estimates (mean and variability) of TSI, persistent entropy, cvTSI and cvTSI n as functions of sample size based on 100 simulations per point. To conclude, this experiment highlights a fundamental distinctio… view at source ↗
Figure 11
Figure 11. Figure 11: Random sampling on intertwined circles using the Alpha complex. (A) Per￾sistence diagram corresponding to the point cloud of [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Effect of noise on TSI, TSigI, entropy, and cvTSI. 4.3. Time series: geometric Brownian motion. We now consider synthetic time series generated by geometric Brownian motion (GBM). Using Takens embedding (the interested readers is referred to [19, 20]), we reconstruct a point cloud from each time series and compute the associated persistence diagrams (for an illustration see Figures 13 and 14). (a) Realiza… view at source ↗
Figure 13
Figure 13. Figure 13: (A) A simulated example of geometric Brownian motion with drift and volatil￾ity (µ, σ) = (0, 0.01) (B) 3 dimensional Takens embedding of the curve with delay parameter τ = 3. (C) Corresponding persistence diagram of Alpha filtration [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: TSI, persistent entropy, and cvTSI as functions of volatility (A) and drift (B). Each value is averaged over 100 Monte Carlo simulations. The persistence diagrams of GBM exhibit a large number of short-lived features, reflecting the absence of strong underlying topological structure. This is consistent with the stochastic and memoryless nature of the process. To assess sensitivity to model parameters, we … view at source ↗
read the original 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 R\'enyi 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 R\'enyi 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper introduces the Topological Stability Index (TSI) as a variance-based scalar summary of persistence lifetimes in barcodes, claiming to establish its scaling behavior, translation invariance, and explicit update formulas under bar insertion/deletion. It also defines a complementary first-moment index (TSigI) and a normalized scale-invariant variant cvTSI, asserting that cvTSI is an affine function of the collision probability ∑p_i² and hence a monotone reparametrization of the Rényi entropy of order two. Numerical experiments on unspecified synthetic data and stochastic time series are presented to argue that TSI captures structural variability complementary to persistent entropy.

Significance. The explicit algebraic identity relating cvTSI to the collision probability (and thus to Rényi-2 entropy) is a direct, parameter-free link between variance-based and entropy-based topological summaries; this is a genuine strength if the supporting derivations are supplied. The claimed invariance and update formulas, if rigorously established, would make TSI practically useful for tracking topological changes in streaming or time-series data where absolute scale matters. The normalization step for cvTSI is intentional and does not undermine the entropy relation, contrary to the stress-test concern.

minor comments (2)
  1. The abstract asserts that fundamental properties, scaling behavior, invariance, and explicit update formulas are established, yet the provided text gives no indication of where the derivations appear (e.g., which section or proposition).
  2. Experimental claims rest on 'synthetic data and stochastic time series' whose generation procedures, barcode extraction methods, and quantitative comparison metrics are not described in the visible text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation of minor revision. The referee correctly identifies the algebraic link between cvTSI and the Rényi-2 collision probability as a key strength, along with the invariance and update formulas. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central algebraic claim—that cvTSI equals an affine transformation of the collision probability ∑p_i²—follows immediately from the definition of cvTSI as the squared coefficient of variation of the lifetime vector after normalization (var(l)/mean(l)² = n∑p_i² − 1). This is an explicit identity derived from the variance formula and the change of variables p_i = l_i/S; it does not rely on fitting, self-citation, or any prior result that itself depends on the target relation. No other load-bearing steps in the provided abstract or description reduce by construction to their inputs. The derivation remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction applies standard definitions of variance and normalized probabilities to barcode lifetimes; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Persistence lifetimes admit a discrete probability distribution after normalization to unit sum.
    Required for both the variance definition of TSI and the algebraic relation to collision probability.

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Reference graph

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