Efficient Geometric Machine Learning for Brain Functional Connectivity Analysis

1924 

Abstract Submission 

Kisung You¹, Hae-Jeong Park²

¹City University of New York, New York, NY, ²Yonsei Univ, Seoul, Seoul

Kisung You  
City University of New York
New York, NY

Hae-Jeong Park  
Yonsei Univ
Seoul, Seoul

The enhanced availability of large-scale neuroimaging data has laid the foundation for population-level analysis of human brain functional connectivity, where brain network is conceptualized as a correlation matrix among signals from regions of interest. This paradigm, considering functional networks as a unit of analysis for inference, has gained popularity in recent years. Traditional approaches to the analysis of a population of functional networks involve concatenating every element in the lower triangular part of each correlation matrix into a vector. While convenient, this method overlooks the intrinsic association structure among edges of functional networks, failing to respect the perspective of brain as a highly correlation object.

In response to the intrinsic limitation to naive vectorization, a line of research has addressed the problem by leveraging a mathematical discipline of Riemannian geometry on the space of functional networks. This perspective treats each correlation matrix as a point on a geometric space known as the correlation manifold. Previous studies have explored the plausibility of a specific geometry, the quotient affine metric, applied to functional networks. Although mathematically elegant, this geometry has a practical drawback that even the simplest task of calculating the distance between two correlation matrices involves solving a nonlinear optimization problem, rendering it impractical for extrapolation into a broader class of problems.

Recently, two alternative geometries - the Euclidean-Cholesky metric and log-Euclidean-Cholesky metric - have been proposed as alternatives due to their computational efficiency and theoretical soundness. These novel geometries employ diffeomorphism from the correlation manifold onto the Euclidean space, allowing for easily optimized computation. We investigate the comparative performance of these novel geometric formulations against established tools, demonstrating that this approach allows for efficient and robust computation while preserving geometric characterization on the relevant space.

Building upon these findings, we introduce a comprehensive toolkit for statistical and machine learning on the space of correlation-based functional networks. Along the development of algorithms, we present theoretical justifications on why our choice of adaptation to the correlation manifold indeed makes sense. Major categories of inferential tasks include computing summary statistics, measuring similarity between two sets of observations, hypothesis testing, cluster analysis, dimensionality reduction for visualization, and regression on scalar responses.

The proposed framework is evaluated on some public datasets including Human Connectome Project (HCP) and ADHD-200. In all experiments, functional networks were defined as partial correlation matrices of the processed functional magnetic resonance imaging signals. To summarize the results, we ran comparative analysis of performances for inferential tasks to predict scalar-valued survey outcomes, identify subtypes of a population, and visualize the distribution of a population of functional networks. In every scenario, we observed that the newly proposed geometries show comparable performance to the incumbent quotient geometry with statistically significant decrease in computational costs.

Our study contributes to the evolving landscape of population-level analysis of human brain functional connectivity by utilizing Riemannian geometry on the correlation manifold. Two geometric structures were introduced to extend our toolbox for machine learning to exploit the computational advantages of the contending geometries. Our contribution adds theoretical justification of geometry-aware inferential methods. Application to the real datasets shows that the new tools makes tremendous reduction in computational resources while maintaining compatible performance metrics at the comparable level.

Classification and Predictive Modeling

Connectivity (eg. functional, effective, structural) ²

Methods Development ¹

Multivariate Approaches

Other Methods

Data analysis

FUNCTIONAL MRI

Machine Learning

Open-Source Software