https://github.com/hugo-strang/silhouette-upper-bound
An upper bound of the Average Silhouette Width.
https://github.com/hugo-strang/silhouette-upper-bound
cluster-analysis clustering clustering-evaluation data-mining data-science machine-learning python python3 silhouette-coefficient silhouette-score upper-bound
Last synced: 6 months ago
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An upper bound of the Average Silhouette Width.
- Host: GitHub
- URL: https://github.com/hugo-strang/silhouette-upper-bound
- Owner: hugo-strang
- License: mit
- Created: 2025-05-28T20:54:01.000Z (about 1 year ago)
- Default Branch: main
- Last Pushed: 2025-11-06T16:15:43.000Z (7 months ago)
- Last Synced: 2025-11-06T18:16:51.029Z (7 months ago)
- Topics: cluster-analysis, clustering, clustering-evaluation, data-mining, data-science, machine-learning, python, python3, silhouette-coefficient, silhouette-score, upper-bound
- Language: Jupyter Notebook
- Homepage: https://pypi.org/project/silhouette-upper-bound/
- Size: 884 KB
- Stars: 5
- Watchers: 0
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# Silhouette Upper Bound
An upper bound of the [Average Silhouette Width](https://en.wikipedia.org/wiki/Silhouette_(clustering)).
*Figure 1: Kmeans clustering applied to a synthetic dataset. Code available [`here`](./experiments/figure_silhouette_samples.py).*
*Figure 2: ASW for varying K. Code available [`here`](./experiments/figure_asw_vs_k.py).*
## Overview
Evaluating clustering quality is a fundamental task in cluster analysis, and the
[Average Silhouette Width](https://en.wikipedia.org/wiki/Silhouette_(clustering)) (ASW) is one of the most widely used metrics for this purpose. ASW scores range from $-1$ to $1$, where:
* Values near 1 indicate well-separated, compact clusters
* Values around 0 suggest overlapping or ambiguous cluster assignments
* Values near -1 imply that many points may have been misassigned
Optimizing the Silhouette score is a common objective in clustering workflows. However, since we rarely know the true global ASW-maximum achievable for a dataset, it's difficult to assess how close a given clustering result is to being globally optimal. Simply comparing to the theoretical maximum of 1 is often misleading, as the structure of the dataset imposes inherent limits on what is achievable.
This project introduces a data-dependent upper bound on the ASW that hopefully can provide a more meaningful reference point than the fixed value of 1. The upper bound helps answer a key question: How close is my clustering result to the best possible outcome on this specific data?
To compute the upper bound, the method requires a dissimilarity matrix as input.
You can find more details in this arXiv [preprint](https://arxiv.org/abs/2509.08625).
## Use Cases
The proposed data-dependent upper bound on the Average Silhouette Width (ASW) opens up opportunities for both research and industry applications.
### Research (Academic Endeavors)
**Confirming global optimality:** Certify that an empirically obtained clustering is within $\epsilon$ of the true ASW maximum.
**Sharper comparisons across algorithms:** Interpret algorithm performance relative to dataset-specific ceilings, rather than the generic $[-1,1]$ scale.
### Industry (Practical Applications)
**Early stopping in optimization loops:** Halt ASW-based searches once solutions are provably close to optimal, saving time and resources.
**Proxy for clusterability:** A low bound indicates limited potential for meaningful clusters, guiding analysts before heavy computation.
**Outlier detection:** Pointwise upper bounds flag observations that cannot fit well into any cluster.
**Constraint-aware clustering:** Incorporate application constraints (e.g. minimal cluster size) via restricted bounds.
### Extension to other clustering quality measures
In practical scenarios where clusters are imbalanced and small groups also matter, the so called *macro-averaged silhouette* (see [this article](https://arxiv.org/abs/2401.05831)) might be more favorable compared to the standard ASW. The macro-silhouette averages first at the cluster level and then across clusters.
In this project we implement an upper bound of this silhouette variant in a solution space that is constrained by fixed cluster sizes.
## Installation
```
pip install silhouette-upper-bound
```
## Examples
To help you get started, we provide example scripts demonstrating common use cases.
You can find these in the [`demos/`](./demos) folder.
## Quickstart
```python
import numpy as np
from silhouette_upper_bound import upper_bound
if __name__ == '__main__':
np.random.seed(42)
# dummy data
A = np.random.rand(100, 100)
D = (A + A.T) / 2
np.fill_diagonal(D, 0)
# ASW upper bound
ub = upper_bound(D)
print(f"There is no clustering of the data points of D that has a higher Silhouette score than {ub}.")
```
## Experimental results
We evaluate the performance of the upper bound using synthetic datasets generated with `scikit-learn`’s `make_blobs()` [function](https://scikit-learn.org/stable/modules/generated/sklearn.datasets.make_blobs.html). Each dataset is identified by a label of the form `n_samples`-`n_features`-`centers`-`cluster_std`, which corresponds to the parameters used in the data generation.
The code that generates the results below can be found in
[`experiments/`](./experiments/table_asw_synthetic_data.py).
| Dataset | KMeans ASW | ASW upper bound | Worst-case relative error |
| --- | --- | --- | --- |
| 400-64-5-6 | 0.249 | 0.376 | 0.38 |
| 400-64-2-2 | 0.673 | 0.673 | .00 |
| 400-128-7-3 | 0.522 | 0.566 | 0.08 |
| 1000-161-2-13 | 0.084 | 0.182 | 0.54 |
Note that the upper bound confirms global optimality for KMeans on dataset 400-64-2-2.
More comprehensive results on synthetic datasets are available in [`results/`](./results/).
## Contribution
Contributions are welcome! If you have suggestions for improvements, bug reports, or new features, feel free to open an issue or submit a pull request.
To contribute:
1. Fork the repository.
2. Create a new branch for your feature or fix.
3. Submit a pull request.
Thank you for helping improve this project!