Open Access
2024 High-dimensional functional graphical model structure learning via neighborhood selection approach
Boxin Zhao, Percy S. Zhai, Y. Samuel Wang, Mladen Kolar
Author Affiliations +
Electron. J. Statist. 18(1): 1042-1129 (2024). DOI: 10.1214/24-EJS2219


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.

Funding Statement

The research of MK is supported in part by NSF Grant ECCS-2216912.


The authors would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper. We would also like to thank Zhaohan Wu from Florida State University for his suggestions on fMRI data analysis. This work was completed in part with resources provided by the University of Chicago Booth Mercury Computing Cluster.


Download Citation

Boxin Zhao. Percy S. Zhai. Y. Samuel Wang. Mladen Kolar. "High-dimensional functional graphical model structure learning via neighborhood selection approach." Electron. J. Statist. 18 (1) 1042 - 1129, 2024.


Received: 1 October 2022; Published: 2024
First available in Project Euclid: 5 March 2024

arXiv: 2105.02487
Digital Object Identifier: 10.1214/24-EJS2219

Primary: 62H22 , 62J07
Secondary: 62P10

Keywords: fMRI data , Functional graphical model , neighborhood selection

Vol.18 • No. 1 • 2024
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