Abstract
Deep learning is having a profound impact on industry and scientific research. Yet, while this paradigm continues to show impressive performance in a wide variety of applications, its mathematical foundations are far from being well established. In this talk, I will present recent developments in this area by illustrating two case studies.
First, motivated by applications in cognitive science, I will present “rating impossibility" theorems. They identify frameworks where deep learning is provably unable to generalize outside the training set for the seemingly simple task of learning identity effects, i.e. classifying whether pairs of objects are identical or not.
Second, motivated by applications in scientific computing, I will illustrate “practical existence" theorems. They combine universal approximation results for deep neural networks with compressed sensing and high-dimensional polynomial approximation theory. As a result, they yield sufficient conditions on the network architecture, the training strategy, and the number of samples able to guarantee accurate approximation of smooth functions of many variables.
Time permitting, I will also discuss work in progress and open questions.
Short bio
Simone Brugiapaglia is Assistant Professor of Mathematics and Statistics at Concordia University. He studied mathematics at the University of Pisa, where he received his BSc in 2010 and his MSc in 2012 and where he held two scholarships from the Istituto Italiano di Alta Matematica. He received his PhD in Mathematical Models and Methods in Engineering from Politecnico di Milano in 2016, where he held a scholarship from the Italian Ministry of Education, Universities and Research. He was a post-doctoral fellow at Ecole polytechnique fédérale de Lausanne in Spring 2016 and at Simon Fraser University from 2016 to 2019. He held a Postdoctoral Training Centre in Stochastic Fellowship from the Pacific Institute for the Mathematical Sciences from 2016 to 2018. In 2019, he was awarded a Leslie Fox Prize for Numerical Analysis by the Institute of Mathematics and its Applications. He has co-authored one book and more than twenty scientific publications, including book chapters, peer-reviewed journal articles, and conference proceedings. His work has been published in venues such as Foundations of Computational Mathematics, Numerische Mathematik, Mathematics of Computation, Neural Computation, and IEEE Transactions on Information Theory. His research interests include mathematics of data science, computational mathematics, and numerical analysis.