Learning from trajectories of positive-definite matrices

Learning from trajectories of positive-definite matrices

Samuel Veer Singh, Trinity College Dublin.

1st of October 2026

Abstract: 

Resting-state functional MRI is often summarised by sliding-window covariance matrices between brain regions, so that each participant is represented by a trajectory of symmetric positive-definite (SPD) matrices over the scan. Such data fall between two toolkits. Functional Data Analysis (FDA) models whole curves but assumes Euclidean values, whereas SPD matrices form a curved space in which naive averages and differences are distorted. Conversely, neural networks for SPD matrices respect this geometry but treat the time points as independent snapshots. We propose a representation-learning framework that combines the two. Geometric layers reduce the SPD matrices while respecting their geometry, and a functional neural network then learns from each whole trajectory, with weight functions that show which parts of the scan matter most. We illustrate the approach on simulations and public clinical neuroimaging data.