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

The paper introduces STRAND, a novel framework that treats persistence diagrams as survival data to simultaneously provide a statistically rigorous non-parametric hypothesis test, interpretable effect sizes, and a stable feature vector for machine learning, all derived from a single coherent representation.

Original authors: Juliette Murris, Bernadette Stolz, Karsten Borgwardt

Published 2026-06-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Juliette Murris, Bernadette Stolz, Karsten Borgwardt

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a bag of marbles. Some are big, some are small, some are red, some are blue. In the world of Topological Data Analysis (TDA), scientists use a tool called "Persistent Homology" to look at data shapes (like a cloud of points or a network of connections) and turn them into these "bags of marbles."

Each marble represents a feature of the shape:

A connected dot (like a single island).
A loop (like a ring or a donut hole).
A void (like the empty space inside a bubble).

For every feature, the tool records two numbers: Birth (when it first appears as you zoom in) and Death (when it disappears). The difference between these two numbers is called Persistence. A feature that lasts a long time is a "strong" signal (like a real donut hole). A feature that appears and vanishes instantly is usually just "noise" (like a speck of dust).

The Problem: Two Separate Toolkits

Until now, scientists had a split personality problem when dealing with these bags of marbles:

The Statisticians wanted to ask: "Are Group A's marbles different from Group B's?" They had tests to answer this, but they couldn't tell you how much different they were, or which specific marbles were the culprits.
The Machine Learning Engineers wanted to feed these marbles into a computer to make predictions (like "Is this protein a drug?"). They needed to turn the bag of marbles into a fixed list of numbers (a vector). But the tools they used to do this didn't help with the statistical questions.

It was like having one tool to measure the height of a tree and a completely different tool to guess its age, with no way to connect the two.

The Solution: STRAND (Survival Topological Representation ANalysis of Diagrams)

The authors introduce STRAND, a new method that treats these topological features like patients in a hospital survival study.

The Creative Analogy: The "Feature Hospital"

Imagine every topological feature (every marble) is a patient entering a hospital.

Birth Time: The moment the patient is admitted.
Death Time: The moment the patient leaves (or passes away).
Persistence: The total time the patient stayed in the hospital.

STRAND asks a simple question: "How long do the patients from Group A stay in the hospital compared to Group B?"

Instead of just counting marbles, STRAND builds a Survival Curve. This is a graph that answers: "What percentage of features are still 'alive' (persisting) after time $t$ ?"

What STRAND Does (The Three Superpowers)

1. The "Fair Judge" (Hypothesis Testing)
STRAND uses a classic medical test called the Log-Rank Test (usually used to compare drug survival rates) to compare the two groups.

The Result: It gives a clear "Yes/No" answer: "Are these two groups of shapes statistically different?"
The Bonus: Unlike older methods, it works even if you only have a tiny number of samples (like just two diagrams).

2. The "Detective" (Interpretable Effect Sizes)
Old methods just gave a "p-value" (a number saying "it's different"). STRAND gives you the story behind the difference.

It tells you the Hazard Ratio: "Features in Group B die 15% faster than in Group A."
It tells you the Median Difference: "The average feature in Group A lasts 3 seconds longer."
It shows you exactly where on the timeline the groups split apart. You can see if the difference happens in the "short-lived" features (noise) or the "long-lived" ones (real structure).

3. The "Translator" (Vectorisation for AI)
To let computers learn from this, STRAND takes that Survival Curve and chops it up into a fixed list of numbers (a vector).

The Magic: This list of numbers is derived from the exact same curve used for the statistical test.
The Benefit: The features the computer uses to make a prediction are the same features the statistician tested for significance. There is no disconnect.
Efficiency: It creates a very short list of numbers (e.g., 50 numbers) compared to other methods that might create thousands, making it much faster for computers to process.

Real-World Checks (What the Paper Actually Found)

The authors tested STRAND in three ways:

Synthetic Shapes: They created fake data (like 4D donuts and spheres) with known shapes and added noise. STRAND correctly identified the differences and controlled for false alarms.
Classification Benchmarks: They tested STRAND on 14 different datasets (molecules, social networks, 3D shapes). STRAND performed just as well as the best existing methods at classifying these shapes, but with much less data and faster computation.
Brain Connectivity (Autism Study): They applied STRAND to brain scan data comparing children and adults.
- Finding: They found that the "loops" (H1 features) in children's brain networks had a different "survival time" than adults'.
- Insight: STRAND didn't just say "they are different"; it quantified that the loops in children's brains "died" (disappeared) faster than in adults, providing a measurable biological difference.

Summary

STRAND bridges the gap between asking a question ("Are these shapes different?") and answering it with a prediction ("Here is a number to feed into an AI"). It does this by viewing data shapes through the lens of survival analysis, turning abstract topological features into a story about how long they last, allowing scientists to measure, compare, and predict with a single, coherent tool.

Technical Summary: From Persistence to Survival (STRAND)

Problem Statement

Topological Data Analysis (TDA) utilizes persistent homology to extract shape-based descriptors from data, outputting persistence diagrams (PDs). While PDs effectively capture topological features (connected components, loops, voids) across scales, they present two significant challenges for practitioners:

Lack of Confirmatory Statistics: Existing methods for comparing PDs are largely descriptive. There is a scarcity of principled hypothesis tests that can confirm topological differences between groups with calibrated Type I error and high statistical power, particularly when sample sizes (number of diagrams) are small.
Incompatibility with Machine Learning: PDs are multisets of varying cardinality, making them incompatible with standard machine learning algorithms that require fixed-dimensional vector inputs.
Disconnected Toolkits: Current solutions address these problems separately. Vectorisation methods (e.g., Persistence Images, Landscapes) enable classification but lack statistical inference capabilities. Conversely, hypothesis testing methods (e.g., permutation tests on Wasserstein distances) yield p-values but fail to provide interpretable effect sizes, localisation of differences, or a unified representation for downstream prediction. This forces practitioners to use separate tools for inference and prediction, breaking the interpretive link between significant features and predictive features.

Methodology: STRAND

The authors introduce STRAND (Survival Topological Representation ANalysis of Diagrams), a framework that unifies hypothesis testing and vectorisation by mapping persistence diagrams to survival data.

Core Concept: Persistence as Survival

STRAND treats the persistence value $p = d - b$ of each topological feature (where $b$ is birth and $d$ is death) as a fully observed "time-to-event."

The Persistence Survival Function: The central object is defined as $S(t) = P(p > t)$ , representing the probability that a randomly chosen feature persists beyond filtration scale $t$ .
Estimation: For a collection of diagrams, STRAND employs the Kaplan-Meier estimator to compute the empirical survival function $\hat{S}(t)$ and the Nelson-Aalen estimator for the cumulative hazard. It accommodates the "delayed-entry" structure of PDs (features are only observable between birth and death) via left truncation.

Key Components

Hypothesis Testing (Section 3.2):
- STRAND compares two groups of PDs by testing the null hypothesis $H_0: S_A(t) = S_B(t)$ against the alternative that they differ for some $t$ .
- It utilizes the log-rank test, a non-parametric statistic comparing observed versus expected events under the null.
- Effect Sizes: Unlike methods yielding only p-values, STRAND provides interpretable effect sizes, including hazard ratios (under proportional hazards assumptions) and median persistence differences ( $\Delta_{med}$ ), along with bootstrap confidence intervals. It also localizes differences across the persistence axis using Kaplan-Meier curves.
Vectorisation (Section 3.3):
- The same survival function $\hat{S}(t)$ used for testing is discretized on a fixed grid to produce a fixed-dimensional feature vector.
- Stability: The resulting vector is proven to be 1-Wasserstein-stable. Theorem 3.1 establishes that the distance between STRAND vectors is bounded by the 1-Wasserstein distance between the input diagrams, ensuring robustness to noise and small perturbations.
- Efficiency: The method produces low-dimensional vectors (e.g., 50–100 features) compared to high-dimensional alternatives like Persistence Images, reducing computational costs for downstream classifiers.

Key Results

1. Statistical Validation (Synthetic Manifolds)

Type I Error Control: Experiments on synthetic manifolds (4D torus, Klein bottle, 3D sphere) with controlled noise show that STRAND maintains calibrated Type I error rates consistent with the nominal $\alpha = 0.05$ , even in the presence of within-diagram correlations (elder rule).
Statistical Power: STRAND demonstrates high power from very small sample sizes. On 4D torus data, it achieves >90% power with as few as two diagrams per group by leveraging the abundance of topological features within each diagram.
Signal Detection: The method successfully distinguishes signal (manifold) from noise features, with log-rank p-values consistently below 0.05 across signal probability regimes.

2. Classification Benchmarks

Performance: STRAND was evaluated on 14 datasets (10 graph datasets from TU Dortmund and 4 3D point cloud datasets).
- On graph datasets, STRAND achieved competitive accuracy (average 61.5%), comparable to state-of-the-art vectorisations like Persistence Images (PI) and Sliced Wasserstein Kernel (SWK).
- On 3D point cloud datasets, STRAND achieved the highest average accuracy (79.5%), outperforming PI (76.7%) and other baselines, with significant margins on complex multi-class datasets.
Computational Efficiency: STRAND offers a massive reduction in feature dimensionality (30–85 $\times$ lower than Persistence Images) and computational time. For large datasets (e.g., ModelNet10), kernel-based methods became infeasible due to $O(N^2)$ complexity, whereas STRAND remained tractable.

3. Case Study: Autism Brain Connectivity

Applied to the ABIDE dataset to compare functional brain connectivity between children (<13) and adults ( $\ge$ 18).
STRAND detected significant developmental differences in loop structures ( $H_1$ ) across three brain atlases, quantifying the effect with hazard ratios ($HR > 1$, indicating shorter persistence in children).
Unlike baseline methods that only provided p-values, STRAND revealed that the magnitude of the effect (HR) varied with parcellation resolution, providing a nuanced, interpretable characterization of topological development.

Significance and Claims

The paper claims that STRAND is the first method to provide hypothesis testing, interpretable effect sizes, and stable vectorisation for persistence diagrams from a single coherent representation.

Unification: It bridges the gap between statistical inference and machine learning, ensuring that the features driving prediction are the same ones tested for significance.
Interpretability: By leveraging survival analysis machinery, STRAND moves beyond scalar p-values to offer localisation of differences (via Kaplan-Meier curves) and quantification of effect magnitude (via hazard ratios and median differences).
Practical Utility: The method is particularly valuable in settings requiring confirmatory studies with small sample sizes (e.g., drug discovery, biomedical imaging) where understanding the magnitude and direction of topological differences is as critical as their existence.

The authors remain modest regarding the proportional hazards assumption, noting that while the log-rank test remains valid without it, the interpretation of the hazard ratio as a single summary effect requires this assumption. In cases of non-proportional hazards, they recommend reporting median persistence differences alongside the survival curves.