Title: Advancing Functional Data Analysis with Deep Learning

Abstract: Functional data are realizations of random functions varying over a continuum, such as images, signals, or spatial-temporal processes. In many modern fields—including neuroscience, medical science, and traffic monitoring—these observations are naturally represented as complex functional data, often characterized by multivariate random functions spanning multiple dimensions. Traditional statistical methods frequently struggle to capture the intricate dependencies among these functional processes. In contrast, deep learning has emerged as a powerful tool for modeling such complex interdependencies without imposing restrictive distributional assumptions. In this talk, I will discuss two key tasks in functional data analysis: functional classification and functional graphical modeling. We propose nonparametric deep learning frameworks for both tasks and establish model consistency for the proposed algorithms. Our theoretical results show that the convergence rates to the true models achieve the classical nonparametric rate, up to a logarithmic factor. Moreover, we identify a novel critical sampling frequency that governs the convergence behavior of deep neural network estimators for both tasks. To demonstrate the effectiveness of our approach, we conduct numerous simulation studies and apply our methods to real-world datasets, including ADNI and ADHD, demonstrating their empirical effectiveness.