Scalable Geometric Learning with Correlation-Based Functional Brain Networks

TL;DR

A novel geometric framework that uses diffeomorphic transformations to embed correlation matrices into a Euclidean space while preserving critical manifold characteristics is proposed, enabling scalable, geometry-aware analyses and integrates seamlessly with standard machine learning techniques, including regression, dimensionality reduction, and clustering.

Abstract

The correlation matrix is a central representation of functional brain networks in neuroimaging. Traditional analyses often treat pairwise interactions independently in a Euclidean setting, overlooking the intrinsic geometry of correlation matrices. While earlier attempts have embraced the quotient geometry of the correlation manifold, they remain limited by computational inefficiency and numerical instability, particularly in high-dimensional contexts. This paper presents a novel geometric framework that employs diffeomorphic transformations to embed correlation matrices into a Euclidean space, preserving salient manifold properties and enabling large-scale analyses. The proposed method integrates with established learning algorithms - regression, dimensionality reduction, and clustering - and extends naturally to population-level inference of brain networks. Simulation studies demonstrate both improved computational speed and enhanced accuracy compared to conventional manifold-based approaches. Moreover, applications in real neuroimaging scenarios illustrate the framework's utility, enhancing behavior score prediction, subject fingerprinting in resting-state fMRI, and hypothesis testing in electroencephalogram data. An open-source MATLAB toolbox is provided to facilitate broader adoption and advance the application of correlation geometry in functional brain network research.

Figures (9)

• Figure 1: Visualization of the $2\times 2$ symmetric and positive-definite (SPD) manifold as the interior of the open upper cone in $\mathbb{R}^3$. The dashed gray lines indicate the coordinate axes, and the red line represents the correlation manifold, with its endpoints excluded, embedded within the SPD region.
• Figure 2: Diagram of the transformation process for ECM and LEC geometries. Applying the mapping $\Theta$ to a full-rank correlation matrix (left) results in a lower-triangular matrix with unit diagonals (middle). The subsequent application of the matrix logarithm to $\mathcal{L}_1^n$ produces strictly lower-triangular matrices with zero diagonals (right).
• Figure 3: Comparison of average wall-clock runtime over 50 trials for computing the distance between perturbed versions of two model correlation matrices, $C_1$ and $C_2$, at varying dimensions $n=10, 20, \ldots, 100$. The $y$-axis represents the average runtime in seconds, displayed on a base-10 logarithmic scale.
• Figure 4: Simulation results for the Fréchet mean computation. For varying dimensions, the Fréchet mean was estimated for a random sample of 100 correlation matrices drawn from the Wishart distribution. Performance under different geometries is reported in terms of (A) the magnitude of error for the estimated Fréchet mean, measured using the Frobenius norm, and (B) the elapsed computation time, measured in seconds.
• Figure 5: Simulation results for the Fréchet median example. For each contamination level, a random sample composed of two types of correlation matrices is drawn, and the Fréchet median is computed. The performance of different geometries is shown in terms of (A) the error magnitude, measured as the Frobenius norm difference between the estimated Fréchet median and the identity matrix, and (B) elapsed computation time, measured in seconds.

Theorems & Definitions (4)

• Definition 1
• Definition 2
• Proposition 4.1
• proof