Mathematical Colloquium Introduction to Mathematics of Artificial Neural Networks (ANNs)

The talk will consist of two parts. In the first part, we introduce Artificial Neural Networks (ANNs) in the context of classical approximation theory with additional insight from neural networks in the brain. We next explain how ANNs work using a simple example of classification of handwritten digits. Historically the development of ANNs was done using empirical methods.  In the second part of the talk, we demonstrate how well-developed mathematical theories and classical theorems can be used to understand and improve the performance of ANNs. Specifically, for several types of ANNs, we use spectral results from Random Matrix Theory and  Fixed Point Theorems for stochastic operators. We conclude by formulating existence and stability results for fixed points of autoencoders. Here we present numerical results supported by rigorous mathematical proofs.