# High-dimensional Functional Graphical Model Structure Learning via Neighborhood Selection Approach

arXiv: https://arxiv.org/abs/2105.02487 · HTML: https://mlsanity.com/paper/2105.02487/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 66 references that are in the map.
- First posted on arXiv 6 May 2021.

## Builds on (most cited first, 9 of 9)

- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression](https://mlsanity.com/paper/1804.02605/) (2018)
- [Conditional Independence Testing in Hilbert Spaces with Applications to Functional Data Analysis](https://mlsanity.com/paper/2101.07108/) (2021)
- [Bayesian Graphical Models for Multivariate Functional Data](https://mlsanity.com/paper/1411.4158/) (2014)
- [Direct Estimation of Differential Functional Graphical Models](https://mlsanity.com/paper/1910.09701/) (2019)
- [Two-sample inference for high-dimensional Markov networks](https://mlsanity.com/paper/1905.00466/) (2019)
- [Partial Separability and Functional Graphical Models for Multivariate Gaussian Processes](https://mlsanity.com/paper/1910.03134/) (2019)
- [Nonparametric and high-dimensional functional graphical models](https://mlsanity.com/paper/2103.10568/) (2021)
- [A test of weak separability for multi-way functional data, with application to brain connectivity studies](https://mlsanity.com/paper/1703.10210/) (2017)

## Built on by (most cited first, 7 of 7)

- [Recursive Causal Discovery](https://mlsanity.com/paper/2403.09300/) (2024)
- [Latent Multimodal Functional Graphical Model Estimation](https://mlsanity.com/paper/2210.17237/) (2022)
- [Trans-Glasso: A Transfer Learning Approach to Precision Matrix Estimation](https://mlsanity.com/paper/2411.15624/) (2024)
- [A Copula Graphical Model for Multi-Attribute Data using Optimal Transport](https://mlsanity.com/paper/2404.06735/) (2024)
- [Differentiable Structure Learning and Causal Discovery for General Binary Data](https://mlsanity.com/paper/2509.21658/) (2025)
- [Learning Functional Graphs with Nonlinear Sufficient Dimension Reduction](https://mlsanity.com/paper/2601.15696/) (2026)
- [High-dimensional Gaussian Graphical Model Testing for Long-Memory Time Series](https://mlsanity.com/paper/2609.30565/) (2026)

## Abstract (arXiv)

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.