Recorded 17 October 2024. Mauro Maggioni of Johns Hopkins University presents "On exploiting compositional structure: one bit of theory and one application" at IPAM's Theory and Practice of Deep Learning Workshop. Abstract: This talk is divided in two parts, one focused on theory and one on applications. In the first part, I will discuss a new, simple model of high-dimensional functions that generalize the single-index model to the case where the linear inner function projecting onto the one-dimensional line spanned by the index vector is replaced by a nonlinear counterpart, given by projection onto a(n unknown) one-dimensional curve. We construct estimators for the regression function that, under suitable assumptions, defeat the curse of dimensionality, achieve nearly optimal learning rates, and can be computed efficiently. In the second part of the talk, I will discuss two recent applications of deep learning in the context of heart digital twins: the first one is the development of a new risk prediction model given clinical and imaging data of patients; the second one is the development of a learning architecture for predicting solutions of parametric PDEs on a family of diffeomorphic domains, which we apply to the prediction of medically-relevant electrophysiological features of heart digital twins. Learn more online at: https://www.ipam.ucla.edu/programs/workshops/workshop-ii-theory-and-practice-of-deep-learning/?tab=overview