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 An Introduction to Fuzzy Sets, Pedrycz W., Gomide F., The MIT Press). Classic logic sets are based on the law of excluded middle that, in a clustering scenario, can be expressed by saying that a sample x_i can belong only to a single cluster c_j. Speaking more generally, if we split our universe into labeled partitions, a hard clustering approach will assign a label to each sample, while a fuzzy (or soft) approach allows managing a membership degree (in Gaussian mixtures, this is an actual probability), w_ij which expresses how strong the relationship is between sample x_i and cluster c_j. Contrary to other methods, by employing fuzzy logic it's possible to define asymmetric sets...