High-dimensional Functional Graphical Model Structure Learning via Neighborhood Selection Approach
This work addresses the challenge of modeling functional data in fields like neuroscience, offering a computationally efficient method, though it is incremental as it builds on existing functional graphical models with a novel estimation technique.
The authors tackled the problem of learning conditional independence structures in high-dimensional functional data, such as EEG and fMRI, by proposing a neighborhood selection approach that estimates the graph directly without requiring a well-defined precision operator, achieving statistical consistency in high-dimensional settings as supported by theory and experiments.
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.