Speaker: Alessandro Scagliotti (TU Munchen)
Title: Control-theoretic approach for the approximation of the optimal transport map

Abstract: In this presentation, we tackle the problem of reconstructing the optimal transport map $T$ between two absolutely continuous measures $\mu,\nu \in \mathcal{P}(\mathbb{R}^n)$, and for this approximation we employ flows generated by linear-control systems in $\mathbb{R}^n$.
We first show that, under suitable assumptions on the measures $\mu,\nu$ and on the controlled vector fields, the optimal transport map is contained in the $C^0_c$-closure of the flows generable by the system.
In the case that discrete approximations $\mu_N,\nu_N$ of the measures $\mu,\nu$ are available, we use a discrete optimal transport plan to set up an optimal control problem. With a $\Gamma$-convergence argument, we prove that its solutions corresponds to flows that provide approximations of the optimal transport map $T$.
Finally, in virtue of the Pontryagin Maximum Principle, we propose an iterative numerical scheme for the resolution of the optimal control problem, resulting in an algorithm for the practical computation of approximations of the optimal transport map. This approach can be interpreted as the construction of a ``Normalizing Flow'' by means of a Residual Neural Network (ResNet). Based on a joint work with Sara Farinelli.

[1] A. Scagliotti, S. Farinelli. Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs
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