Topology meets classification.
Investigating whether topological features from persistent homology complement classical morphological features for breast cancer classification on the WDBC dataset.
1.0Abstract
This independent study asks whether the shape of data — captured through persistent homology — adds useful information to the classical morphological features commonly used for breast cancer classification. Using the Wisconsin Diagnostic Breast Cancer dataset (569 observations, 30 features), local point clouds were built in a PCA-reduced space, Vietoris–Rips filtrations were computed, and H0 and H1 persistence diagrams were vectorised as persistence images. Classical (30), topological (50) and combined (80) feature sets were evaluated with Logistic Regression, an RBF Support Vector Machine and a Random Forest under stratified five-fold cross-validation. Classical features performed strongest overall; topological features offered an alternative but weaker representation; and the combined set produced some model-specific improvements without consistently outperforming the classical baseline.
2.0Problem
Breast cancer classification models typically rely on morphological measurements of cell nuclei. These features are highly informative, but they describe each measurement independently. Topological Data Analysis offers a complementary lens: it summarises how observations relate to one another geometrically, at many scales at once. The open question is whether that structural signal adds value on top of the features already in use.
3.0Objectives
- Construct topological features from the WDBC data using persistent homology.
- Compare classical, topological and combined feature representations on equal terms.
- Evaluate three model families with a consistent, stratified cross-validation protocol.
- Report results transparently, including where topological features did not help.
4.0Dataset
- Name
- Wisconsin Diagnostic Breast Cancer (WDBC)
- Observations
- 569
- Features
- 30
A public benchmark dataset of morphological features computed from digitised images of breast mass cell nuclei.
5.0Methodology
- 5.1
Principal Component Analysis
Reduce the 30 morphological features to a lower-dimensional space.
- 5.2
Local point-cloud construction
Build a local neighbourhood point cloud around each observation.
- 5.3
Vietoris–Rips complexes
Grow simplicial complexes across a range of distance scales.
- 5.4
Persistent homology
Track connected components (H0) and loops (H1) across the filtration.
- 5.5
Persistence images
Vectorise persistence diagrams into 50 fixed-length topological features.
- 5.6
Classification
Logistic Regression, RBF SVM and Random Forest on each feature set.
6.0Experimental setup
- Feature sets: Classical (30), Topological (50), Combined (80).
- Models: Logistic Regression, RBF Support Vector Machine, Random Forest.
- Validation: stratified five-fold cross-validation.
- Metrics: ROC AUC and F1, reported as mean ± variation across folds.
Classical30
Morphological features from the WDBC dataset.
Topological50
Persistence-image features from H0 and H1.
Combined80
Classical and topological features together.
7.0Results
- Classical · 30
- Topological · 50
- Combined · 80
Logistic Regression
RBF SVM
Random Forest
- Classical · 30
- Topological · 50
- Combined · 80
Logistic Regression
RBF SVM
Random Forest
| AUC · Feature set | Logistic Regression | RBF SVM | Random Forest |
|---|---|---|---|
| Classical(30) | 0.9953 ± 0.0053 | 0.9945 ± 0.0060 | 0.9885 ± 0.0083 |
| Topological(50) | 0.9482 ± 0.0233 | 0.9280 ± 0.0256 | 0.9402 ± 0.0170 |
| Combined(80) | 0.9951 ± 0.0057 | 0.9921 ± 0.0068 | 0.9915 ± 0.0038 |
Highlighted: highest mean per model. Differences are not tested for statistical significance.
| F1 · Feature set | Logistic Regression | RBF SVM | Random Forest |
|---|---|---|---|
| Classical(30) | 0.9633 ± 0.0240 | 0.9688 ± 0.0222 | 0.9381 ± 0.0186 |
| Topological(50) | 0.8638 ± 0.0313 | 0.8433 ± 0.0257 | 0.8443 ± 0.0403 |
| Combined(80) | 0.9641 ± 0.0202 | 0.9311 ± 0.0310 | 0.9264 ± 0.0243 |
Highlighted: highest mean per model. Differences are not tested for statistical significance.
8.0Findings
- Classical features performed strongest overall, with the highest mean AUC (0.9953, Logistic Regression) and the highest mean F1 (0.9688, RBF SVM).
- Topological features alone provided an alternative representation of the data but performed less strongly in this experiment (AUC 0.928–0.948).
- Combined features showed some model-specific improvements — for example Random Forest AUC rose from 0.9885 to 0.9915 — but did not consistently outperform classical features, and RBF SVM F1 decreased.
9.0Limitations
- A single, well-studied public dataset; results have not been externally validated on independent data.
- No statistical significance testing is claimed for the differences between feature sets.
- Not clinically validated, and not intended for diagnostic use.
- Topological features depend on construction choices (PCA dimension, neighbourhood size, filtration and image parameters) that were not exhaustively explored.
10.0Future work
- Evaluate on independent breast cancer datasets to test generalisation.
- Apply formal statistical comparison of feature representations.
- Run sensitivity analyses on topological construction parameters.
- Compare alternative vectorisations such as persistence landscapes and Betti curves.