High-dimensional Functional Graphical Model Structure Learning via Neighborhood Selection Approach
Boxin Zhao, Percy S. Zhai, Y. Samuel Wang, Mladen Kolar
TL;DR
This paper develops a neighborhood selection framework for high-dimensional Gaussian functional graphical models by framing graph learning as a sequence of function-on-function regressions and solving a finite-dimensional vector-on-vector regression with group LASSO penalties. It avoids relying on a population precision operator, enabling true infinite-dimensional applicability, and provides non-asymptotic guarantees for neighborhood and graph recovery. The methodology allows flexible basis choices, with practical guidance favoring FPCA-based representations, and supports parallel computation via ADMM. Empirical results from simulations and fMRI data (ASD and ADHD) show improved graph recovery over functional graphical lasso and related methods, illustrating the approach’s robustness and potential for neuroimaging applications.
Abstract
Undirected graphical models are widely used to model the conditional independence structure of vector-valued data. However, in many modern applications, for example those involving EEG and fMRI data, observations are more appropriately modeled as multivariate random functions rather than vectors. Functional graphical models have been proposed to model the conditional independence structure of such functional data. We propose a neighborhood selection approach to estimate the structure of Gaussian functional graphical models, where we first estimate the neighborhood of each node via a function-on-function regression and subsequently recover the entire graph structure by combining the estimated neighborhoods. Our approach only requires assumptions on the conditional distributions of random functions, and we estimate the conditional independence structure directly. We thus circumvent the need for a well-defined precision operator that may not exist when the functions are infinite dimensional. Additionally, the neighborhood selection approach is computationally efficient and can be easily parallelized. The statistical consistency of the proposed method in the high-dimensional setting is supported by both theory and experimental results. In addition, we study the effect of the choice of the function basis used for dimensionality reduction in an intermediate step. We give a heuristic criterion for choosing a function basis and motivate two practically useful choices, which we justify by both theory and experiments.