I have been examining the literature on optimal transport, and typically it is introduced as an arbitrary number of probability measures $\mu_0, \mu_1, \cdots$ on an n-dimensional manifold $N$.
The situation I wish to examine is a lot simpler - optimal transport between two point sets $\vec{p}_i, \vec{q}_j$ in $\mathbb{R}^2$, where the probability measures for both of these point sets is taken to be a uniform histogram. That is to say, the distribution of the points in $\mathbb{R}^2$ can be arbitrary, but the weights of the points are uniform.
The number of points in each set are given by $N_1, N_2$. The papers I have read indicate that optimal transport algorithms between point sets can be executed even when $N_1 \neq N_2$.
This fact is really not making sense to me - if optimal transport takes you continuously from one point distribution to another, and the number of points varies between distributions, doesn't this mean that at some point during the transport some of the points must vanish/reappear? This seems like a discontinuous operation that doesn't fit within the overall context of optimal transport.