@mi2018variational], we can alternatively solve for $\bm{\Region}$ and $\yL{\IdxCentroid} = 1/|\RegionL{\IdxCentroid}|\sum_{\x \in \RegionL{\IdxCentroid}} \x$. The energy  will monotonically decrease and eventually converge into a cycle of one. Figure \[fig:regularized\_kmeans\] illustrates the regularized K-means result which informally looks like an interpolation between K-means and constrained K-means.

![Results from different regularization strength $\lambda$ in . Left is traditional K-means and right is constrained K-means.[]{data-label="fig:regularized_kmeans"}](regularized_kmeans.pdf){width="\columnwidth"}

Co-clustering Spatial Features in $\mathbb{R}^{n}$ {#sec:co_cluster}
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Extending the Wasserstein clustering procedure to multiple targets induces the co-clustering problem. In this section, we discuss the connection between co-clustering problems and VWBs. In particular, because we use quadratic Euclidean distances as the ground metric, we focus on co-clustering spatial features embedded in the Euclidean space.

Given multiple distributional data, the goal of co-clustering is to simultaneously partition each domain to 1) minimize the pairwise variance in the same cluster and 2) minimize the pairwise variance for each cluster across domains. We assume all samples reside in $\mathbb{R}^{n}$ equipped with $\|\cdot\|_{2}$, then: $$\begin{split}
    \underset{\RRegionL{i}}{\min}\ \EnergyL{9}[\RRegionL{i}] \eqdef & \sum_{i=1}^{\N} \sum_{\IdxCentroid=1}^{\K} \frac{1}{2|\RegionL{i,\IdxCentroid}|} \sum_{\x,\x' \in \RegionL{i,\IdxCentroid}} \|\x - \x'\|_{2}^{2} \\
    + \sum_{\IdxCentroid=1}^{\K}  \sum_{1 \leq i < j \leq \N} & \frac{\lambda_{i,j,\IdxCentroid}}{|\RegionL{i,\IdxCentroid}|+|\RegionL{j,\IdxCentroid}|} \sum_{\substack{\x \in \RegionL{i,\IdxCentroid} \\ \x' \in \RegionL{j,\IdxCentroid} }} \|\x - \x'\|_{2}^{2}. \nonumber
