Functional connectomes are symmetric positive definite matrices, which means they live on a curved Riemannian manifold rather than in flat Euclidean space. Most standard statistical tools assume flatness, so applying them directly produces distorted distances — the frequently used analogy is measuring the distance between two cities with a straight ruler while ignoring the curvature of the Earth.
Geometry-aware methods take that curvature into account. Tangent space analysis is the most common approach: pick a reference point on the manifold, use the Riemannian logarithmic map to project nearby connectomes into the flat tangent space at that point, and then apply ordinary linear methods — regression, PCA, distance metrics — to the projections.
Two choices dominate whether this works in practice: which reference point you project around, and how much regularization you apply to keep near-singular matrices away from the boundary of the manifold. Both involve real trade-offs, including a data-leakage risk in the reference choice.
CNS Projects¶
Davor Curic — applying tangent space analysis to track Alzheimer’s disease progression through the movement of functional connectomes over time. His page carries the most detailed technical writeup in this book, covering the choice of reference and the bias-variance trade-off in regularization — start there rather than here
Getting started¶
Davor Curic’s technical description — the substantive treatment
fMRI — where the connectomes come from
Alzheimer’s Disease — the topic hub