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arxiv: 2606.11911 · v1 · pith:QKFWZOCGnew · submitted 2026-06-10 · 📊 stat.ML · cs.LG· math.AT

From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Features

Juliette Murris , Bernadette Stolz , Karsten Borgwardt This is my paper

Pith reviewed 2026-06-27 08:16 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.AT
keywords persistence diagramssurvival analysistopological data analysishypothesis testingeffect sizesvectorizationmachine learningnonparametric statistics
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The pith

Treating persistence values in diagrams as survival times produces a unified representation for statistical tests, effect sizes, and stable feature vectors.

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

The paper presents STRAND, which reframes collections of persistence diagrams by treating each feature's persistence p = d - b as a fully observed time-to-event observation. The central object becomes the persistence survival function S(t) = P(p > t), from which the authors derive a non-parametric two-sample test, interpretable effect sizes, and a feature vector stable under the 1-Wasserstein distance. This single representation is intended to connect hypothesis testing of topological features directly to downstream machine learning tasks without separate pipelines. A reader would care if the approach delivers calibrated error rates and competitive performance on graph and point-cloud benchmarks while remaining interpretable.

Core claim

STRAND treats persistence diagrams as survival data, with each topological feature's persistence serving as a time-to-event datum and the survival function S(t) = P(p > t) as the central object; from this representation the method derives a non-parametric two-sample test with calibrated Type I error, interpretable effect sizes, and a 1-Wasserstein-stable feature vector for machine learning.

What carries the argument

The persistence survival function S(t) = P(p > t), which encodes the distribution of persistence values across diagrams and serves as the source for both the test statistic and the feature vector.

If this is right

  • A non-parametric test becomes available that controls Type I error and achieves high power even with few diagrams.
  • Effect sizes derived from the survival function supply an interpretable measure of difference between diagram collections.
  • The resulting feature vector remains stable under 1-Wasserstein perturbations and can be fed directly into standard machine learning pipelines.
  • The same representation applies to functional brain connectivity analysis in fMRI data.

Where Pith is reading between the lines

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

  • The survival framing may allow direct borrowing of censoring or competing-risks techniques when some topological features are only partially observed.
  • Because the vector is 1-Wasserstein stable, it could be combined with existing Wasserstein-based kernels without additional stability proofs.
  • The approach might generalize to other topological summaries that admit a natural ordering or magnitude, such as persistence landscapes or silhouettes.

Load-bearing premise

Treating each topological feature's persistence value as a fully observed time-to-event datum preserves all information needed for valid statistical comparison and stable vectorization.

What would settle it

Running the proposed two-sample test on synthetic manifolds with controlled topology differences and checking whether the empirical Type I error rate stays at the nominal level (for example 0.05) across repeated small-sample draws would directly test the calibration claim.

Figures

Figures reproduced from arXiv: 2606.11911 by Bernadette Stolz, Juliette Murris, Karsten Borgwardt.

Figure 1
Figure 1. Figure 1: From a point cloud (here, a torus in 1), persistent homology produces a persistence diagram [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Hypothesis testing validation on 4D torus (500 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Bridging topological data and survival analysis fields with [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Kaplan-Meier estimators of topological features in 4D torus data with signal parameter of [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Signal vs. noise discrimination analysis across Klein bottle, and spheres with four type of [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Power vs hazard ratio for STRAND two-sample test on simulated torus data. Left: H0 features. Right: H1 features. Color indicates signal probability. The V-shaped curve confirms test validity: power ≈ 0 at HR = 1 (no true difference) and power → 1 as | log(HR)| increases [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
read the original abstract

Persistence diagrams are common representations in topological data analysis, but they do not naturally live in a vector space, and the statistical tools developed for comparing them have largely evolved separately from those used for downstream prediction. We introduce STRAND (Survival Topological Representation ANalysis of Diagrams), which treats (collections of) PDs as survival data: each topological feature with persistence value $p = d - b$ is a fully observed time-to-event, and the persistence survival function $S(t) = \mathbb{P}(p > t)$ is the central object for comparing diagrams. From this single representation we derive (i) a non-parametric two-sample test with calibrated Type I error and high power from a small number of diagrams; (ii) interpretable effect sizes; and (iii) a 1-Wasserstein-stable feature vector for downstream machine learning. We validate calibration and power on synthetic manifolds with controlled topology, demonstrate competitive vectorisation across 14 graph and 3D point cloud benchmarks, and apply the method to study functional brain connectivity in fMRI/neuroscience data. To our knowledge, STRAND is the first method to provide hypothesis testing and vectorisation for persistence diagrams from a single coherent and interpretable representation.

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

2 major / 1 minor

Summary. The manuscript introduces STRAND, which models collections of persistence diagrams as survival data by treating each topological feature's persistence value p = d - b as a fully observed time-to-event observation. The persistence survival function S(t) = P(p > t) is positioned as the central object, from which the authors derive (i) a non-parametric two-sample test with calibrated Type I error and high power, (ii) interpretable effect sizes, and (iii) a 1-Wasserstein-stable feature vector for downstream machine learning. The approach is validated on synthetic manifolds with controlled topology, 14 graph and 3-D point cloud benchmarks, and an fMRI neuroscience application, with the claim that it is the first method to unify hypothesis testing and vectorization from a single coherent representation.

Significance. If the representation proves sufficient for both valid inference and stable vectorization, the work would provide a unified, interpretable bridge between topological data analysis and survival analysis tools, potentially simplifying statistical practice in TDA while offering effect-size interpretability that is currently rare in diagram comparisons.

major comments (2)
  1. [Abstract] Abstract: the claim that the derived feature vector is '1-Wasserstein-stable' does not follow from the construction. The survival function S(t) is computed solely from the multiset of persistence values p = d - b; however, the 1-Wasserstein distance on persistence diagrams is defined via optimal matching in the (b, d) plane (with diagonal projection), so two diagrams can share identical persistence multisets yet exhibit arbitrarily large W1 distance when their birth coordinates differ. This directly challenges the stability assertion for the vectorization component.
  2. [Abstract] Abstract (paragraph on the survival representation): the assumption that the marginal distribution on p is a sufficient statistic for both the hypothesis test and the vectorization is load-bearing but unexamined. Because standard PD distances and stability results operate on the full (b, d) coordinates, treating p-values as i.i.d. time-to-event data risks invalidating the claimed Type I error calibration and the 1-Wasserstein stability when birth times carry topological information.
minor comments (1)
  1. [Abstract] The abstract states 'competitive vectorisation across 14 graph and 3D point cloud benchmarks' without naming the baselines, metrics, or exclusion rules; these details are needed to assess the performance claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the derived feature vector is '1-Wasserstein-stable' does not follow from the construction. The survival function S(t) is computed solely from the multiset of persistence values p = d - b; however, the 1-Wasserstein distance on persistence diagrams is defined via optimal matching in the (b, d) plane (with diagonal projection), so two diagrams can share identical persistence multisets yet exhibit arbitrarily large W1 distance when their birth coordinates differ. This directly challenges the stability assertion for the vectorization component.

    Authors: We agree with the referee's analysis. The feature vector is obtained from the survival function of the persistence values p alone and therefore cannot be guaranteed to satisfy stability with respect to the 1-Wasserstein distance on the full (b, d) diagrams. We will revise the abstract to remove the specific claim of 1-Wasserstein stability. revision: yes

  2. Referee: [Abstract] Abstract (paragraph on the survival representation): the assumption that the marginal distribution on p is a sufficient statistic for both the hypothesis test and the vectorization is load-bearing but unexamined. Because standard PD distances and stability results operate on the full (b, d) coordinates, treating p-values as i.i.d. time-to-event data risks invalidating the claimed Type I error calibration and the 1-Wasserstein stability when birth times carry topological information.

    Authors: The referee correctly notes that the method relies on the marginal distribution of persistence values. While the synthetic experiments demonstrate calibrated Type I error under this modeling choice, birth coordinates can carry additional information in some applications. We will add a clarifying paragraph in the discussion section acknowledging this modeling assumption and its scope, without altering the core procedure. revision: partial

Circularity Check

0 steps flagged

No circularity: STRAND applies standard survival methods to a new representation

full rationale

The paper introduces the survival function S(t) = P(p > t) computed from persistence values p = d - b as an explicit modeling choice, then derives the two-sample test, effect sizes, and 1-Wasserstein-stable vector by applying established non-parametric survival techniques (e.g., log-rank or Kaplan-Meier style estimators) to this representation. No equation reduces any claimed output to a fitted parameter or input quantity by construction, and the provided abstract and description contain no self-citations that serve as load-bearing uniqueness theorems or ansatzes. The derivation chain is therefore self-contained: the statistical properties follow from the external validity of survival analysis once the representation is adopted, rather than from re-labeling or re-fitting the same data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the domain assumption that persistence values behave like survival times; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Persistence values p = d - b can be treated as fully observed time-to-event data for constructing a survival function S(t) = P(p > t)
    This modeling choice is the foundation for deriving the test, effect sizes, and vector from one object (abstract).

pith-pipeline@v0.9.1-grok · 5759 in / 1284 out tokens · 22523 ms · 2026-06-27T08:16:27.812350+00:00 · methodology

discussion (0)

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