Optimal Transport

Discrete Measures Definition: Shannon-Boltzmann entropy Let $P \in U(a, b)$ be a coupling matrix for discrete measures with vectors $a$ and $b$. Then, the Shannon-Boltzmann entropy is $$ H(P) = - \sum_{i, j} P_{i, j} \log{(P_{i, j})}, $$ where $0 = \log{0}$. Note that $$ \nabla^2 H(P) = - \diag(P_{i, j}^{-1}). $$ So, $H$ is strictly concave. Let us add a regularization term to the discrete Kantorovich problem. $\varepsilon$ will be our regularization weight and it works as a kind of “temperature”. ...

Summary Up to this point we have studied Monge and Kantorovich problems. Duality of Kantorovich and the Wasserstein metric. Slicing OT as a way to lower bound $\W_p$. Now, we are interested in solving the Kantorovich problem in an alternative way. The goal here is to find a condition that is sufficient to retrieve a map from $\mathcal{X}$ to $\mathcal{Y}$. Then, the optimal transport will be given by the objects that minimize a certain criterion over this condition. ...