Are there alternative regularizations for optimal transport problems besides entropic regularization? I see that most of the regularization done involves an entropy term.
Has there been any work done on other regularization methods? In particular, I'm wondering if anyone has done a regularization involving a deviation of the induced OT map from some other fixed map. For example, the term may be $\varphi(\nabla\phi,I)$ for the case where the fixed map is the identity. Here, $\nabla\phi$ is the transport map (gradient of a convex function $\phi$) and $\varphi$ is some divergence measure.
 A: Perhaps not exactly what you have in mind: F. Santambrogio, J. Louet, and L. De Pascale have considered a gradient regularization for the Monge problem, i-e
\begin{equation}
\min\limits_{T\#\mu=\nu} \int |T(x)-x| d\mu(x) +\varepsilon \int |\nabla T(x)|^2
\tag{$P_{\varepsilon}$}
\end{equation}
(I'm not writing any measure in the gradient penalization since the choice of such a reference measure matters quite a lot, as one can imagin and as in entropic optimal transport.) One expects that $(P_{\varepsilon})$ converges to the classical transport problem as $\varepsilon \to 0$, but the result is more subtle than meets the eye. See their paper.
A: One type of deviation from another map goes under name of "optimal transport over linear/nonlinear dynamics'. Described in the continuous form of OT (Brenier-Benamou style), here the baseline dynamics are given either by a linear or nonlinear dynamical system, and the job is to find the deviation to baseline that leads to desired transport in optimal way. This idea generalizes optimal control to optional transport of measures. Google scholar with these terms will help you.
