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 $u_{ij}$.

${\sum }_{j=1}^K{\sum }_{i=1}^N{u}_{ij}^f{\Vert x_i-{\mu }_j\Vert }^2\to {\min }_{{\mu }_j,u_{ij}}\text{ s.t. }{\sum }_{j=1}^K{u}_{ij}^f=1\forall i$

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