In the context of this exercise, can the centroids be incorrect even if there is convergence? How?
Yes, absolutely. K-Means clustering finishes once the centroids no longer update, and have converged, even if they are completely incorrect. There are several reasons as to why this can happen for K-Means clustering, which are given as follows.
One common reason is that the value chosen for k was not ideal. A value that is too low can cause centroids to include more than one cluster’s data points, and a value too large can cause clusters to be divided further among different centroids.
Furthermore, there can be incorrect assumptions made about the data. One wrong assumption is that the data is always clustered in spherical shapes, which is not always the case. The data can take on many different shapes, like a ring, or even a long rectangular shape. And, another incorrect assumption is that clusters are the same size, but, you may have some clusters that are much more dense than others or much larger in size.
Despite these possible reasons for having an incorrect result, K-Means clustering attempts to find a solution by repeating the entire process several times and then choosing the best result of those runs. However, this does not resolve all the possible issues, like a wrong choice of k.