Riemannian Geometry for Noise-Robust Covariance Network Analysis of Schizophrenia EEG: Geometric-Entropic Signatures of Dysconnectivity.

Functional brain networks in schizophrenia (SZ) are often characterized by covariance-based measures, yet covariance matrices live on a curved geometric structure rather than in ordinary Euclidean space, complicating noise-robust inference from scalp EEG. We develop a Riemannian Geometry-based Adaptive Nonlinear Coupling Analysis (RGA-NCA) framework that integrates the affine-invariant Riemannian metric (AIRM), tangent space mapping (TSM), and an anatomically adaptive artifact rejection (AAAR) strategy accounting for regional signal-to-noise heterogeneity. The framework is grounded in the observation that Euclidean summaries of symmetric positive definite matrices are sensitive to noise-driven volume inflation, whereas geodesic distances on the manifold emphasize shape deformation. RGA-NCA was evaluated on four benchmark dynamical systems, a supplementary multichannel EEG-like sample covariance simulation, and a public button-tone SZ/HC EEG dataset associated with the auditory feedback paradigm described by Ford et al. (81 subjects; 49 SZ, 32 healthy controls). Compared with Euclidean and linear baselines, RGA-NCA showed lower sensitivity to noise-driven distance distortion and yielded clearer group-level contrasts in the tested ROI analyses; all four pre-specified frontotemporal and parietal channel pairs remained significant after Benjamini-Hochberg FDR correction. The resulting patterns are consistent with reduced long-range connectivity together with localized hyper-synchronization-like effects in SZ. Quantitatively, the Riemannian structural sensitivity index (sim=exp(-d2/4)) remained high across all tested SNR levels (-20 to +10 dB; 50 Monte Carlo trials per level; range 0.936-0.964), with only a 0.026 endpoint change between +10 and -20 dB, whereas the Euclidean metric fell from 0.922 at +10 dB to 0.000 at -20 dB. These findings support Riemannian modeling as a candidate strategy for noisy covariance-based neural data, pending validation in larger independent cohorts.