0$) would minimize the first term in . Thus, we can directly give the solution to as $\{\frac{\lambda}{1 + \lambda}\hL{\IdxCentroid}\}_{\IdxCentroid=1}^{\K}$.

**Case 2:** $\NUL{\IdxCentroid} \in (0,1),\ \sum_{\IdxCentroid = 1}^{\K} \NUL{\IdxCentroid} = 1$. It is also trivial to verify that minimizing the second term in  over $\bm{\Region}$ yields $w_{\RegionL{\IdxCentroid}} = \NUL{\IdxCentroid}w$ (replace $\frac{1}{K}$ with $\NUL{\IdxCentroid}$). $w_{\RegionL{\IdxCentroid}} = \NUL{\IdxCentroid}w$ also triggers the convergence of VOT as in Case 1.

At this point, we claim that VOT, , minimizes the total transportation cost regardless of the measures equal or not. We leave rigorous proofs to future work. We illustrate the convergence in Figure \[fig:uot\]. The top half shows VOT between balanced measures and the bottom half shows unbalanced measures, $w = 900 = 500 + 2 \times 200, \NUL{\IdxCentroid} = \frac{1}{3}$. Note that the gradient of the VOT and VWB, , correlates to the absolute measure values. Thus, we should scale the step size, $\eta$ in , for each VOT according to the difference of the measure, i.e. $\eta_i / w $, assuming the total for $\NU$ is $1$. Figure \[fig:uot\] shows that under the same (scaled) GD step size, VOT in two cases follows the same trend.

We apply VWBs to unbalanced measures 