Clustering instantaneous phase-locking states with a rank-two Bures-Wasserstein distance

Abstract

Instantaneous phase-locking (iPL) matrices, defined by the cosines of pairwise phase differences between band-limited signals, underpin widely used methods for time-resolved functional connectivity, including Leading Eigenvector Dynamics Analysis (LEiDA). Each iPL matrix has rank at most two and is fully characterized by its two nonzero eigenvalues and corresponding eigenvectors. Yet, existing methods retain different parts of this decomposition and compare them using distances that do not reflect the geometry of the underlying sample space. We show that weighting the two eigenvectors by the square roots of their eigenvalues recovers each channel's phase relative to the mean phase, and that optimal orthogonal Procrustes alignment of these weighted eigenvector pairs yields the Bures-Wasserstein distance between the corresponding iPL matrices. In EEG, where most channels oscillate nearly in phase, existing distances become insensitive to phase-lag size, whereas the Bures-Wasserstein distance preserves this sensitivity. In simulations, clustering with this distance recovered states differing in both lag size and spatial pattern that existing distances failed to distinguish.In resting-state EEG, it resolved distinct phase-gradient states that other methods merged into a single global in-phase state.