\in \RegionL{i,\IdxCentroid}} \left[\x - \left[\sum_{j=1}^{\N}  \alpha_{j, \IdxCentroid}\right] \right]^2 + C \\
    \eqdef &     \sum_{i=1}^{\N}  \sum_{\IdxCentroid=1}^{\K} \sum_{\x \in \RegionL{i,\IdxCentroid}} \left(\x - \hat{\alpha_{\IdxCentroid}} \right)^2 + C.
\end{split}$$ $C$ is some constant. Thus, we convert co-clustering to $N$ $K$-means problems with the same set of centroids.

Then, we can naturally impose a constraint on the weights, i.e $\int_{\RegionL{i, \IdxCentroid}}d\MUL{i}(\x) = \NUL{\IdxCentroid},\ \forall i,\ \IdxCentroid$, to turn the problem into a VWB problem which is also an $N$ constrained K-means problem. Note, that it is trivial to extend it into a generalized VWB problem, by instead inserting the weighted constraint into the main objective as we did in \[sec:regularized\_kmeans\].

### Regularized VWBs for Co-Clustering {#sec:regularized_vwb}

In addition to purely clustering feature domains according to Wasserstein losses, we can regularize the correspondences based on prior knowledge. Inspired by [@alvarezmelis2019towards; @mi2018regularized], we regularize the correspondence by global invariances. Directly regularizing Monge correspondences is highly intractable because Monge maps are basically binary permutations and thus not differentiable. Therefore, we instead regularize the centroid update process.

To this end, instead of using the average of the centroids as we did in , we estimate the rigid transformation (isometry) between the VWB and the centroids of each domain by 