Fuzzy K-Means Clustering

Fuzzy k-means Clustering in contrast to normal Clustering will not find discrete centroids for each datapoint but rather a probability distribution over all clusters uiju_{ij}.

∑j=1K∑i=1Nuijf∥xi−μj∥2→min⁡μj,uij s.t. ∑j=1Kuijf=1∀i\sum_{j=1}^K \sum_{i=1}^N u_{i j}^f\left\|x_i-\mu_j\right\|^2 \rightarrow \min _{\mu_j, u_{i j}} \text { s.t. } \sum_{j=1}^K u_{i j}^f=1 \forall i

We can use the fuzzifier ff to control how discrete or not the clusters are.