I have a question about calculating the partition matrix for the Fuzzy C-Means (FCM) Clustering Algorithm. For any point $x_i$ and cluster centroid $c_j$, the membership value $w_{i,j}$ is computed by the following algorithm (where c is the number of clusters, m is a fuzziness hyper-parameter, and $\Vert \Vert$ is Euclidean distance): $$w_{i,j}=\sum_{k=1}^c \frac{1}{\left(\frac{\Vert x_i-c_j\Vert}{\Vert x_i-c_k\Vert}\right)^{\frac{2}{m-1}}}$$ Theoretically (although very unlikely experimentally), any point could have a distance of $0$ from any centroid, causing a division by $0$.
The solution seems obvious to me: if $\Vert x_i-c_k\Vert=0$, then point $x_i$ lies directly on centroid $c_k$, so $w_{i,k}=1$ and $w_{i,j}=0$ for all other j, preserving the requirement that $\sum_{j=1}^c w_{i,j}=1$, but I'm not sure if this is sound according to the algorithm.
If point $x_i$ lies on centroid $c_j$, is $w_{i,j}=1$ true?
(Just looking for some verification, I couldn't find anything in the source materials I was viewing...)
This is a special case of the theorem where it is assumed that no $c_k=x_i$.
The original paper this formula appeared in is:
The article can be found her:
and the theorem is Theorem 3, (a) Case 1 on page 44.