This talk explores tools from functional analysis that explain the exceptional performance of deep neural networks, which form the backbone of most state-of-the-art artificial intelligence systems. Taking center stage in this discussion is a relatively new mathematical framework that precisely details the functional properties of trained neural networks. The methodology for this framework relies heavily on transform-domain sparse regularization, the Radon transform of computed tomography, and approximation theory.

Furthermore, the framework clarifies several key aspects of neural network training and architecture. It provides insights into the effect of weight decay regularization in training, the relevance of skip connections and low-rank weight matrices in network design, the significance of sparsity in neural networks, and the proficiency of neural networks in handling high-dimensional problems. Moreover, the characterization of neural functions within this framework reveals new insights into the complexity and structure of deep neural networks. This fresh perspective opens a novel pathway to effectively study the behavioral traits of deep networks.