Log-Euclidean Frameworks for Smooth Brain Connectivity Trajectories

2025-07-21

AI Summary

To address trajectory non-smoothness, geometric distortion, and the generation of non-positive-definite correlation matrices in dynamic functional connectivity (dFC) modeling, this paper proposes a smoothing framework grounded in the Log-Euclidean Riemannian geometry. First, time-varying correlation matrices are mapped to Euclidean space via the matrix logarithm. Second, polynomial regression is performed in this Euclidean domain—enabling differentiable and interpretable trajectory fitting for the first time. Finally, the fitted trajectory is mapped back via the matrix exponential, rigorously ensuring that every reconstructed point is a valid, positive-definite correlation matrix. The method guarantees geometric consistency with the manifold structure of correlation matrices, achieves computational efficiency through closed-form Euclidean operations, and preserves neurobiological interpretability. In sliding-window dFC reconstruction, it significantly enhances trajectory stability and fidelity to underlying dynamic evolution patterns.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

The brain is often studied from a network perspective, where functional activity is assessed using functional Magnetic Resonance Imaging (fMRI) to estimate connectivity between predefined neuronal regions. Functional connectivity can be represented by correlation matrices computed over time, where each matrix captures the Pearson correlation between the mean fMRI signals of different regions within a sliding window. We introduce several Log-Euclidean Riemannian framework for constructing smooth approximations of functional brain connectivity trajectories. Representing dynamic functional connectivity as time series of full-rank correlation matrices, we leverage recent theoretical Log-Euclidean diffeomorphisms to map these trajectories in practice into Euclidean spaces where polynomial regression becomes feasible. Pulling back the regressed curve ensures that each estimated point remains a valid correlation matrix, enabling a smooth, interpretable, and geometrically consistent approximation of the original brain connectivity dynamics. Experiments on fMRI-derived connectivity trajectories demonstrate the geometric consistency and computational efficiency of our approach.

Problem

• Modeling smooth brain connectivity trajectories from fMRI data
• Mapping dynamic connectivity into Euclidean space for regression
• Ensuring geometric consistency in connectivity matrix approximations

Innovation

• Log-Euclidean Riemannian framework for smooth connectivity
• Mapping trajectories to Euclidean spaces via diffeomorphisms
• Polynomial regression on transformed correlation matrices