Can silhouette be calculated with distances to centroids, instead of pairwise point distances?

Question

I am using Silhouette cluster validation for each repetition (for a specific K) of k-means, k-modes and k-medoids.

All the definitions of Silhouette I see calculate the distance of each point to others points within the same cluster, then compare it to the distance of the same point with points in other clusters, in turn.

From wikipedia:

$ a(i) = $ average distance between point $i$ and other points in the same cluster

$ b(i) = $ average distance between point $i$ and all points in the closest cluster

$ s(i) = \frac{b(i) - a(i)}{\max\{a(i),b(i)\}} $

I would expect similar results by, instead, comparing each point with its own centroid, and then the centroids of other clusters, in turn.

Is this a good approach, or do I risk losing information? It's much more efficient computationally (with my dataset, it's a concern).

Answer 1

If your distance function is squared Euclidean, then you can exploit additivity and Konig-Huygens.

Variance is equivalent to the average pairwise squared deviation. Similarly, the squared deviation from the center of another cluster should be the average squared deviation (but not the squared average) to the cluster members.

The same trick will not work with other distances.