Geometric learning of functional brain network on the correlation manifold

Dynamic and low-dimensional modeling of brain functional connectivity on Riemannian manifolds

Modeling brain functional connectivity (FC) is key in investigating brain functions and dysfunctions. FC is typically quantified by symmetric positive definite (SPD) matrices that are located on a Riemannian manifold rather than the regular Euclidean space, whose modeling faces three challenges. First, FC can be time-varying and the temporal dynamics of FC matrix time-series need to be modeled within the constraint of the SPD Riemannian manifold geometry, which remains elusive. Second, the FC matrix time-series exhibits considerable stochasticity, whose probability distribution is difficult to model on the Riemannian manifold. Third, FC matrices are high-dimensional and dimensionality reduction methods for SPD matrix time-series are still lacking. Here, we develop a Riemannian state-space modeling framework to simultaneously address the challenges. First, we construct a new Riemannian state-space model (RSSM) to define a hidden SPD matrix state to achieve dynamic, stochastic, and low-dimensional modeling of FC matrix time-series on the SPD Riemannian manifold. Second, we develop a new Riemannian Particle Filter (RPF) algorithm to estimate the hidden low-dimensional SPD matrix state and predict the FC matrix time-series. Third, we develop a new Riemannian Expectation Maximization (REM) algorithm to fit the RSSM parameters. We evaluate the proposed RSSM, RPF, and REM using simulation and real-world EEG datasets, demonstrating that the RSSM enables accurate prediction of the EEG FC time-series and classification of emotional states, outperforming traditional Euclidean methods. Our results have implications for modeling brain FC on the SPD Riemannian manifold to study various brain functions and dysfunctions.

ℛSCZ: A Riemannian schizophrenia diagnosis framework based on the multiplexity of EEG-based dynamic functional connectivity patterns

Abnormal electrophysiological (EEG) activity has been largely reported in schizophrenia (SCZ). In the last decade, research has focused to the automatic diagnosis of SCZ via the investigation of an EEG aberrant activity and connectivity linked to this mental disorder. These studies followed various preprocessing steps of EEG activity focusing on frequency-dependent functional connectivity brain network (FCBN) construction disregarding the topological dependency among edges. FCBN belongs to a family of symmetric positive definite (SPD) matrices forming the Riemannian manifold. Due to its unique geometric properties, the whole analysis of FCBN can be performed on the Riemannian geometry of the SPD space. The advantage of the analysis of FCBN on the SPD space is that it takes into account all the pairwise interdependencies as a whole. However, only a few studies have adopted a FCBN analysis on the SPD manifold, while no study exists on the analysis of dynamic FCBN (dFCBN) tailored to SCZ. In the present study, I analyzed two open EEG-SCZ datasets under a Riemannian geometry of SPD matrices for the dFCBN analysis proposing also a multiplexity index that quantifies the associations of multi-frequency brainwave patterns. I adopted a machine learning procedure employing a leave-one-subject-out cross-validation (LOSO-CV) using snapshots of dFCBN from (N-1) subjects to train a battery of classifiers. Each classifier operated in the inter-subject dFCBN distances of sample covariance matrices (SCMs) following a rhythm-dependent decision and a multiplex-dependent one. The proposed ℛSCZ decoder supported both the Riemannian geometry of SPD and the multiplexity index DC reaching an absolute accuracy (100 %) in both datasets in the virtual default mode network (DMN) source space.

Classification of BCI-EEG Based on the Augmented Covariance Matrix

OBJECTIVE

Electroencephalography signals are recorded as multidimensional datasets. We propose a new framework based on the augmented covariance that stems from an autoregressive model to improve motor imagery classification.

METHODS

From the autoregressive model can be derived the Yule-Walker equations, which show the emergence of a symmetric positive definite matrix: the augmented covariance matrix. The state-of the art for classifying covariance matrices is based on Riemannian Geometry. A fairly natural idea is therefore to apply this Riemannian Geometry based approach to these augmented covariance matrices. The methodology for creating the augmented covariance matrix shows a natural connection with the delay embedding theorem proposed by Takens for dynamical systems. Such an embedding method is based on the knowledge of two parameters: the delay and the embedding dimension, respectively related to the lag and the order of the autoregressive model. This approach provides new methods to compute the hyper-parameters in addition to standard grid search.

RESULTS

The augmented covariance matrix performed ACMs better than any state-of-the-art methods. We will test our approach on several datasets and several subjects using the MOABB framework, using both within-session and cross-session evaluation.

CONCLUSION

The improvement in results is due to the fact that the augmented covariance matrix incorporates not only spatial but also temporal information. As such, it contains information on the nonlinear components of the signal through the embedding procedure, which allows the leveraging of dynamical systems algorithms.

SIGNIFICANCE

These results extend the concepts and the results of the Riemannian distance based classification algorithm.

Pseudo-online framework for BCI evaluation: a MOABB perspective using various MI and SSVEP datasets

Objective. BCI (Brain-Computer Interfaces) operate in three modes:online,offline, andpseudo-online. Inonlinemode, real-time EEG data is constantly analyzed. Inofflinemode, the signal is acquired and processed afterwards. Thepseudo-onlinemode processes collected data as if they were received in real-time. The main difference is that theofflinemode often analyzes the whole data, while theonlineandpseudo-onlinemodes only analyze data in short time windows.Offlineprocessing tends to be more accurate, whileonlineanalysis is better for therapeutic applications.Pseudo-onlineimplementation approximatesonlineprocessing without real-time constraints. Many BCI studies beingofflineintroduce biases compared to real-life scenarios, impacting classification algorithm performance.Approach. The objective of this research paper is therefore to extend the current MOABB framework, operating inofflinemode, so as to allow a comparison of different algorithms in apseudo-onlinesetting with the use of a technology based on overlapping sliding windows. To do this will require the introduction of a idle state event in the dataset that takes into account all different possibilities that are not task thinking. To validate the performance of the algorithms we will use the normalized Matthews correlation coefficient and the information transfer rate.Main results. We analyzed the state-of-the-art algorithms of the last 15 years over several motor imagery and steady state visually evoked potential multi-subjects datasets, showing the differences between the two approaches from a statistical point of view.Significance. The ability to analyze the performance of different algorithms inofflineandpseudo-onlinemodes will allow the BCI community to obtain more accurate and comprehensive reports regarding the performance of classification algorithms.

R-Mixup: Riemannian Mixup for Biological Networks

Biological networks are commonly used in biomedical and healthcare domains to effectively model the structure of complex biological systems with interactions linking biological entities. However, due to their characteristics of high dimensionality and low sample size, directly applying deep learning models on biological networks usually faces severe overfitting. In this work, we propose R-Mixup, a Mixup-based data augmentation technique that suits the symmetric positive definite (SPD) property of adjacency matrices from biological networks with optimized training efficiency. The interpolation process in R-Mixup leverages the log-Euclidean distance metrics from the Riemannian manifold, effectively addressing the swelling effect and arbitrarily incorrect label issues of vanilla Mixup. We demonstrate the effectiveness of R-Mixup with five real-world biological network datasets on both regression and classification tasks. Besides, we derive a commonly ignored necessary condition for identifying the SPD matrices of biological networks and empirically study its influence on the model performance. The code implementation can be found in Appendix E.