I'm reading Optimal Transport for Domain Adaptation as seen here: https://arxiv.org/abs/1507.00504
We assume that we have source domain $\Omega_S$ with variables $X^s$ and labels $Y^s$, and target domain $\Omega_T$ with variables $X^t$ and labels $Y^t$. We assume there is some transformation $T \colon \Omega_S \to \Omega_T$ and that $P_s(y| x^s) = P_t(y|T(x^s))$.
We can estimate distribution of $X^S$ and $X^T$ empirically, and then calculate the optimal coupling $\gamma_0$.
What I don't get is part 3.3 in the paper.
what does it mean to "interpolate" the two distributions here? That is, why are we even finding $\hat{\mu}$ at all? Also, for $t = 1$, wouldn't $\hat{\mu}$ just be $\mu_t$?
How is $\hat{x_i^s}$ derived here?
Thank you.