Alessandro Scagliotti

TUM Munich

30 novembre 2023, 14.00
Salle X133
Campus Toulon

Ensemble optimal control: ResNets, diffeomorphisms approximation and
Normalizing Flows

In the last years it was observed that Residual Neural
Networks (ResNets) can be interpreted as discretizations of control
systems, bridging ResNets (and, more generally, Deep Learning) with
Control Theory. In the first part of this seminar we formulate the task
of a data-driven reconstruction of a diffeomorphism as an ensemble
optimal control problem. In the second part we adapt this machinery to
address the problem of Normalizing Flows: after observing some samplings
of an unknown probability measure, we want to (approximately) construct
a transport map that brings a “simple” distribution (e.g., a Gaussian)
onto the unknown target distribution. In both the problems we use tools
from $\Gamma$-convergence to study the limiting case when the size of
the data-set tends to infinity.

This talk is based on the papers

Deep Learning approximation of diffeomorphisms via linear-control systems.

Normalizing flows as approximations of optimal transport maps via
linear-control neural ODEs