the kernel k-means problem is notated as:
$ \overset{min}{U \in \{0,1\}^{kxs}} \overset{max}{c_k(\cdot) \in \mathcal{H}_{K}} \sum_{j=1}^{k}\sum_{i=1}^{S} \frac{U_{ji}}{s} \left \| c_j(\cdot)-\kappa(x_i, \cdot)) \right \|^2_{\mathcal{H}_K} $
U is the cluster membership matrix which is 1 if a point is in the cluster and 0 if it is not. $ c_k(\cdot) $ are the cluster centers and $ \kappa (x_i, \cdot) $ is the kernel function (use RBF for example). K are the clusters and S are the points.
It might be an easy question, but could somebody explain in easy words what's going on here?
Usually in k-means we are trying to minimize the distances between the points and the cluster centers (as done with $c_{j}(\cdot) - \kappa(x_i, \cdot) $. Fine, but what is the max doing here? What means that $\overset{max}{c_k(\cdot)}$? Are we searching the maximum cluster center point? Why?
Unfortunately I did not find a youtube video or paper where this formula is explained in detail. Thanks a lot in advance.
Additional info:
The whole formula comes from "Stream Clustering: efficient kernel-based approximation using importance sampling" by Radha Chitta, Jin & Jain 2016 but I think a better and easier introduction is given in "Kernel-based clustering of Big Data" by Radha Chitta 2015 in formula 1.7.
$ \overset{min}{U \in \rho} \overset{max}{c_k(\cdot) \in \mathcal{H}_{K}} \sum_{k=1}^{C}\sum_{i=1}^{n} U_{ki} \left \| c_k(\cdot)-\kappa(x_i, \cdot)) \right \|^2_{\mathcal{H}_K} $
Where U is the cluster membership matrix, which is one if a point is in a cluster, domain $\rho = {U \in {0, 1}^{C×n} : U^T = 1}$, where 1 is a vector of all ones, $\mathcal{H}_{K}$ is the reproducing hilbert space, $c_k(\cdot)$ are the cluster centers, C are the clusters, n are the points.