\end{split}$$ where $|\RegionL{i,\IdxCentroid}|$ is the number of samples in $\RegionL{i,\IdxCentroid}$; $\lambda_{i,j,\IdxCentroid} \in \{0, 1\}$ specifies the correspondence of the clusters across different domains. Thus, $\sum_{i} \lambda_{i,j,\IdxCentroid} = 1$ and $\sum_{j} \lambda_{i,j,\IdxCentroid} = 1$. As for K-means, we can simplify the pairwise variance with the mean of each cluster at each domain, $\alpha_{i, \IdxCentroid}$: $$\label{eq:cocluster}
\begin{split}
    \underset{\RRegionL{i}}{\min}\ \EnergyL{9}[\RRegionL{i}] \equiv & \sum_{i=1}^{\N} \sum_{\IdxCentroid=1}^{\K} \sum_{\x\in \RegionL{i,\IdxCentroid}} \|\x - \alpha_{i, \IdxCentroid}\|_{2}^{2} \\
    + \sum_{\IdxCentroid=1}^{\K} &  \sum_{i = 1}^{\N} \sum_{j \neq i} \lambda_{i,j,\IdxCentroid} \sum_{\x \in \RegionL{i,\IdxCentroid}} \|\x - \alpha_{j, \IdxCentroid}\|_{2}^{2}.
\end{split}$$ where $\alpha_{i, \IdxCentroid} = \frac{1}{|\RegionL{i, \IdxCentroid}|} \sum_{\x \in \RegionL{i, \IdxCentroid}} \x$

is the cluster center for each cluster at each domain. The first term of is solving $\N$ K-means problems. The second term is solving $\N(\N-1)$ K-means problems but with the cluster centroids at other domains. Thus, we can further simplify the problem into: $$\label{eq:cocluster2}
\begin{split}
    \underset{\bm{\RegionL{i}}}{\min}\ \EnergyL{9}[\bm{\RegionL{i}}] \equiv \ignore{&
    \sum_{i=1}^{\N} \sum_{j=1}^{\N} \sum_{\IdxCentroid=1}^{\K} \sum_{\x \in \RegionL{i,\IdxCentroid}} \|\x - \alpha_{j, \IdxCentroid}\|_{2}^{2}\\}
    &     \sum_{i=1}^{\N}  \sum_{\IdxCentroid=1}^{\K} \sum_{\x \in \RegionL{i,\IdxCentroid}} \sum_{j=1}^{\N} \|\x - \alpha_{j, \IdxCentroid}\|_{2}^{2}\\
    \end{split}$$ Solving involves alternatively updating partition $\{\bm{\RegionL{i}}\}_{i}$ and the centroid $\{\alpha_{i, \IdxCentroid}\}_{i, \IdxCentroid}$. When updating $\{\bm{\RegionL{i}}\}_{i}$ with fixed $\{\alpha_{i, \IdxCentroid}\}_{i, \IdxCentroid}$, we can rewrite as $$\label{eq:cocluster3}
\begin{split}
    \EnergyL{11}[\RRegionL{i}]
    = &     \sum_{i=1}^{\N}  \sum_{\IdxCentroid=1}^{\K} \sum_{\x 