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Learning Functional Graphs with Nonlinear Sufficient Dimension Reduction

Kyongwon Kim, Bing Li

TL;DR

This work introduces a nonparametric functional graphical model (f-SGM) that uses nonlinear sufficient dimension reduction to compress high-dimensional conditioning sets when assessing conditional independence among random functions. By embedding the problem in a hierarchy of RKHSs and employing functional GSIR to obtain low-dimensional sufficient predictors $U^{ij}$, followed by a hybrid conjoined conditional covariance operator (CCCO) to test independence, the method relaxes Gaussian and copula assumptions while mitigating the curse of dimensionality. Empirical results on simulated nonlinear and heteroscedastic models show that f-SGM outperforms existing nonparametric and additive methods, and application to an f-MRI ADHD dataset reveals meaningful differences in brain connectivity between ADHD and control groups. The approach is computationally scalable and amenable to parallelization, offering a practical framework for functional network inference in neuroscience and related domains.

Abstract

Functional graphical models have undergone extensive development during the recent years, leading to a variety models such as the functional Gaussian graphical model, the functional copula Gaussian graphical model, the functional Bayesian graphical model, the nonparametric functional additive graphical model, and the conditional functional graphical model. These models rely either on some parametric form of distributions on random functions, or on additive conditional independence, a criterion that is different from probabilistic conditional independence. In this paper we introduce a nonparametric functional graphical model based on functional sufficient dimension reduction. Our method not only relaxes the Gaussian or copula Gaussian assumptions, but also enhances estimation accuracy by avoiding the ``curse of dimensionality''. Moreover, it retains the probabilistic conditional independence as the criterion to determine the absence of edges. By doing simulation study and analysis of the f-MRI dataset, we demonstrate the advantages of our method.

Learning Functional Graphs with Nonlinear Sufficient Dimension Reduction

TL;DR

This work introduces a nonparametric functional graphical model (f-SGM) that uses nonlinear sufficient dimension reduction to compress high-dimensional conditioning sets when assessing conditional independence among random functions. By embedding the problem in a hierarchy of RKHSs and employing functional GSIR to obtain low-dimensional sufficient predictors , followed by a hybrid conjoined conditional covariance operator (CCCO) to test independence, the method relaxes Gaussian and copula assumptions while mitigating the curse of dimensionality. Empirical results on simulated nonlinear and heteroscedastic models show that f-SGM outperforms existing nonparametric and additive methods, and application to an f-MRI ADHD dataset reveals meaningful differences in brain connectivity between ADHD and control groups. The approach is computationally scalable and amenable to parallelization, offering a practical framework for functional network inference in neuroscience and related domains.

Abstract

Functional graphical models have undergone extensive development during the recent years, leading to a variety models such as the functional Gaussian graphical model, the functional copula Gaussian graphical model, the functional Bayesian graphical model, the nonparametric functional additive graphical model, and the conditional functional graphical model. These models rely either on some parametric form of distributions on random functions, or on additive conditional independence, a criterion that is different from probabilistic conditional independence. In this paper we introduce a nonparametric functional graphical model based on functional sufficient dimension reduction. Our method not only relaxes the Gaussian or copula Gaussian assumptions, but also enhances estimation accuracy by avoiding the ``curse of dimensionality''. Moreover, it retains the probabilistic conditional independence as the criterion to determine the absence of edges. By doing simulation study and analysis of the f-MRI dataset, we demonstrate the advantages of our method.
Paper Structure (23 sections, 2 theorems, 63 equations, 9 figures, 5 tables)

This paper contains 23 sections, 2 theorems, 63 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Functional sufficient graphical models
  3. Functional nonlinear sufficient dimension reduction
  4. Determining conditional independence
  5. Sample-level implementation of f-SGM
  6. Coordinate mapping
  7. First-level Hilbert spaces
  8. Second-level Hilbert spaces
  9. f-GSIR
  10. Third-level Hilbert spaces
  11. Hybrid CCCO and its HS-norm
  12. Determination of tuning parameters
  13. Simulation comparisons with existing methods
  14. Nonlinear models
  15. Model II.
  16. ...and 8 more sections

Key Result

Theorem 1

If ${\cal{F}} ^{ -(i,j)}$ is the central $\sigma$-field for $X ^{ (i,j)} | X ^{ -(i,j)}$ for each $(i,j) \in {\mathsf{V}}$, then

Figures (9)

  • Figure 1: Example of human brain network (left) and f-MRI functional data for control group (upper right) and ADHD group (lower right).
  • Figure 2: ROC curves for four estimators. Upper left: balanced case with $n=100$; lower left: balanced case with $n=200$; upper right: unbalanced case with $n=100$; lower right: unbalanced case with $n=200$.
  • Figure 3: ROC curves for four estimators. Upper left: balanced case with $n=100$; lower left: balanced case with $n=200$; upper right: unbalanced case with $n=100$; lower right: unbalanced case with $n=200$.
  • Figure 4: ROC curves for four estimators. Upper left: balanced case with $n=100$; lower left: balanced case with $n=200$; upper right: unbalanced case with $n=100$; lower right: unbalanced case with $n=200$.
  • Figure 5: ROC curves for model $\space \textup{IV}$. Left: $n=100$; right: $n=200$.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Definition 2
  • Theorem 3