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

Overview

Authors: Rui Song¹, Jinhan He¹, Jun Wang²

ORCID iDs: Rui Song, Jinhan He

Smart Health Big Data Analysis and Location Services Engineering Research Center of Jiangsu Province, School of Chemistry and Life Sciences, Nanjing University of Posts and Telecommunications, Nanjing 210023, China; (R.S.); (J.H.)
Smart Health Big Data Analysis and Location Services Engineering Research Center of Jiangsu Province, School of Applied Technology, Nanjing University of Posts and Telecommunications, Nanjing 210023, China

Institutions: Nanjing University of Posts and Telecommunications (China)

Journal: Entropy (Basel, Switzerland), volume 28, issue 6, article 644

Dates: received 19 May 2026; accepted 5 June 2026; published online 8 June 2026

Type: Research article · Language: English

License: CC BY

Identifiers: DOI 10.3390/e28060644 · PMID 42352154 · PMCID PMC13297899 · OpenAlex W7163892576

Open access: gold, a free copy (OpenAlex)

Status: dead link

Categories: EEG (modality), human (organism), schizophrenia / psychosis (population)

Methods: Connectivity, Spectral & time-frequency, Statistics, Smoothing, state filtering, decompositions, Machine learning, Preprocessing, fMRI & imaging, Physiology & signal measures

Keywords: Riemannian geometry, EEG, schizophrenia, covariance manifold, information geometry, entropy, brain connectivity, affine-invariant metric, nonlinear coupling

Topic: Functional Brain Connectivity Studies (Cognitive Neuroscience, Neuroscience), according to OpenAlex

Funding: Science and Technology Innovation Training Program for College Students in Jiangsu Province (202510293017Z)

Citations: not cited yet (Europe PMC); 37 references in the paper

Abstract

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.

Reproduced under the paper's license (CC BY), from the paper cited above.

Code

No file of the authors' code could be read here: it is described below, and read at its source.

kaggle.com/datasets/broach

License: none: the authors keep all their rights

State: the link is dead, verified on 27 September 2026

Evidence: found in the paper

Software Heritage: not checked

Found in: “Data Availability Statement”

Not found: README, license file, CITATION.cff, environment file, tests, continuous integration, documentation

Availability: 1 check, the latest on 27 September 2026: the link is dead (HTTP 404)

27 September 2026: the link is dead (HTTP 404)

At the source: kaggle.com/datasets/broach/button-tone-sz

The paper's code and data availability statement is in the Data section.

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Data

No dataset and no data link were found in the paper.

Data Availability Statement

The raw EEG data that support the findings of this study are openly available in Kaggle at https://www.kaggle.com/datasets/broach/button-tone-sz (accessed on 4 March 2026), as a public button-tone SZ/HC EEG dataset associated with the auditory feedback paradigm described by Ford et al. Derived subject-level feature tables and cleaned analysis scripts supporting the reported group-level statistics are provided in the Supplementary Materials.

Reproduced under the paper's license (CC BY), from the paper cited above.

Versions

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Version 1, 27 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 3 authors, 9 keywords, 1 funder, 31 references.

Cite

This paper

Song, R., He, J., & Wang, J. (2026). Riemannian Geometry for Noise-Robust Covariance Network Analysis of Schizophrenia EEG: Geometric-Entropic Signatures of Dysconnectivity. Entropy (Basel, Switzerland), 28(6), 644. https://doi.org/10.3390/e28060644

BibTeX

@article{song2026riemannian, author = {Song, Rui and He, Jinhan and Wang, Jun}, title = {{Riemannian Geometry for Noise-Robust Covariance Network Analysis of Schizophrenia EEG: Geometric-Entropic Signatures of Dysconnectivity}}, journal = {Entropy (Basel, Switzerland)}, year = {2026}, month = jun, volume = {28}, number = {6}, pages = {644}, publisher = {Multidisciplinary Digital Publishing Institute (MDPI)}, issn = {1099-4300}, doi = {10.3390/e28060644}, url = {https://doi.org/10.3390/e28060644}, pmid = {42352154}, pmcid = {PMC13297899} }

RIS

TY - JOUR AU - Song, Rui AU - He, Jinhan AU - Wang, Jun TI - Riemannian Geometry for Noise-Robust Covariance Network Analysis of Schizophrenia EEG: Geometric-Entropic Signatures of Dysconnectivity T2 - Entropy (Basel, Switzerland) J2 - Entropy (Basel) PY - 2026 DA - 2026/06/08 VL - 28 IS - 6 SP - 644 SN - 1099-4300 PB - Multidisciplinary Digital Publishing Institute (MDPI) DO - 10.3390/e28060644 UR - https://doi.org/10.3390/e28060644 LA - en ER -

CSL-JSON

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