Fuzzy C-means
We have already talked about the difference between hard and soft clustering, comparing K-means with Gaussian mixtures. Another way to address this problem is based on the concept of fuzzy logic, which was proposed for the first time by Lotfi Zadeh in 1965 (for further details, a very good reference is Pedrycz W., Gomide F., An Introduction to Fuzzy Sets, The MIT Press, 1998). Classic logic sets are based on the law of excluded middle, which in a clustering scenario can be expressed by saying that a point can belong only to a single cluster cj.
Speaking more generally, if we split our universe into labeled partitions, a hard clustering approach would assign a label to each sample, while a fuzzy (or soft) approach would allow the management of a membership degree (in Gaussian mixtures, this is an actual probability) wij, which expresses how strong the relationship is between point and cluster cj.
Contrary to other methods, by employing fuzzy logic it's possible...