Deep Neural Network for Functional Graphical Models Structure Learning

Functional data refer to data that are realizations of random functions varying over a continuum, such as images or signals. In many modern fields, including neuroscience, medical science, and traffic monitoring, observations are better modeled as multivariate random functions rather than as vectors. To capture the conditional independence structure of such multivariate functional data, functional graphical models have been developed. In this paper, we propose a novel and flexible method to estimate the neighborhood of each node using a deep neural network-based functional data regression and feature selection approach with an arbitrary nonparametric form. The full graph structure is then recovered by combining the estimated neighborhoods. Our approach avoids common distributional assumptions on the random functions and circumvents the need for a well-defined precision operator, which may not exist in the functional data context. Furthermore, we establish model consistency for the proposed algorithm. The convergence rate reaches to the classical non-parametric regression rate up to a logarithmic factor. We discover a novel critical sampling frequency that governs the convergence rates of the deep neural network estimator for both densely and sparsely observed functional data. The empirical performance of our method is demonstrated through simulation studies and a real data application.