Simone Brugiapaglia,
Department of Mathematics and Statistics,
Concordia University
"The mathematical foundations of deep learning: from rating impossibility to practical existence theorems"
Nov 09, 2022, Schedule:
- Nespresso & Teatime ( 417 DSL - Commons )
- 03:00 to 03:30 PM Eastern Time (US and Canada)
- Colloquium - F2F ( 499 DSL ) / Virtual ( Zoom )
- 03:30 to 04:30 PM Eastern Time (US and Canada)
Meeting # 942 7359 5552
Abstract:
Deep learning is having a profound impact on scientific research. Yet, while deep neural networks continue to show impressive performance in a wide variety of fields, including scientific computing, their mathematical foundations are far from being well established. In this talk, we will present recent developments in this area by illustrating two case studies.
First, motivated by applications in cognitive science, we will present “rating impossibility" theorems. These theorems identify frameworks where neural networks are provably unable to generalize outside the training set while performing the seemingly simple task of learning identity effects, i.e. classifying pairs of objects as identical or not.
Second, motivated by applications in scientific computing, we will illustrate “practical existence" theorems. These theorems 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, we will also discuss work in progress and open questions in this research area.
