Alessandro Scagliotti - Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems

Presentation given by Alessandro Scagliotti on the 6th April 2022 in the one world seminar on the mathematics of machine learning on the topic "Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems".

Abstract: In the last years it has been observed that Residual Neural Networks (ResNets) can be interpreted as discretizations of control systems. This can be a valuable tool for a further mathematical understanding of Machine Learning, since it bridges ResNets (and, more generally, Deep Learning) with Control Theory. This parallelism can be useful to study existing architectures and to develop new ones. In particular, in the present seminar we investigate ResNets obtained from linear-control systems. Despite their simplicity, recent theoretical results guarantee that they could be surprisingly expressive. We will focus on the problem of producing an approximation of a diffeomorphism after observing its action on a finite ensemble of points (the dataset). In this framework, the training of the ResNet corresponds to the resolution of a proper Optimal Control problem. Finally, we will see that, owing to the linear dependence of the system in the controls, the training algorithms based on Pontryagin Maximum Principle can be carried out with low computational effort.

Видео Alessandro Scagliotti - Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems канала One world theoretical machine learning