Silhouette Upper Bound

An upper bound of the Average Silhouette Width.

Overview

Evaluating clustering quality is a fundamental task in cluster analysis, and the Average Silhouette Width (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.

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/ folder.

Quickstart

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. 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/table1.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.

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!