Riemannian Approach Research Articles

A Riemannian geometric approach for timelike and null spacetime geodesics

The geodesic motion in a Lorentzian spacetime can be described by trajectories in a 3-dimensional Riemannian metric. In this article we present a generalized Jacobi metric obtained from projecting a Lorentzian metric over the directions of its Killing vectors. The resulting Riemannian metric inherits the geodesics for asymptotically flat spacetimes including the stationary and axisymmetric ones. The method allows us to find Riemannian metrics in three and two dimensions plus the radial geodesic equation when we project over three different directions. The 3-dimensional Riemannian metric reduces to the Jacobi metric when static, spherically symmetric and asymptotically flat spacetimes are considered. However, it can be calculated for a larger variety of metrics in any number of dimensions. We show that the geodesics of the original spacetime metrics are inherited by the projected Riemannian metric. We calculate the Gaussian and geodesic curvatures of the resulting 2-dimensional metric, we study its near horizon and asymptotic limit. We also show that this technique can be used for studying the violation of the strong cosmic censorship conjecture in the context of general relativity.

Benchmarking brain–computer interface algorithms: Riemannian approaches vs convolutional neural networks

Objective.To date, a comprehensive comparison of Riemannian decoding methods with deep convolutional neural networks for EEG-based brain-computer interfaces remains absent from published work. We address this research gap by using MOABB, The Mother Of All BCI Benchmarks, to compare novel convolutional neural networks to state-of-the-art Riemannian approaches across a broad range of EEG datasets, including motor imagery, P300, and steady-state visual evoked potentials paradigms.Approach.We systematically evaluated the performance of convolutional neural networks, specifically EEGNet, shallow ConvNet, and deep ConvNet, against well-established Riemannian decoding methods using MOABB processing pipelines. This evaluation included within-session, cross-session, and cross-subject methods, to provide a practical analysis of model effectiveness and to find an overall solution that performs well across different experimental settings.Main results.We find no significant differences in decoding performance between convolutional neural networks and Riemannian methods for within-session, cross-session, and cross-subject analyses.Significance.The results show that, when using traditional Brain-Computer Interface paradigms, the choice between CNNs and Riemannian methods may not heavily impact decoding performances in many experimental settings. These findings provide researchers with flexibility in choosing decoding approaches based on factors such as ease of implementation, computational efficiency or individual preferences.

Efficient Feature Learning Model of Motor Imagery EEG Signals with L1-Norm and Weighted Fusion.

Brain-computer interface (BCI) for motor imagery is an advanced technology used in the field of medical rehabilitation. However, due to the poor accuracy of electroencephalogram feature classification, BCI systems often misrecognize user commands. Although many state-of-the-art feature selection methods aim to enhance classification accuracy, they usually overlook the interrelationships between individual features, indirectly impacting the accuracy of feature classification. To overcome this issue, we propose an adaptive feature learning model that employs a Riemannian geometric approach to generate a feature matrix from electroencephalogram signals, serving as the model's input. By integrating the enhanced adaptive L1 penalty and weighted fusion penalty into the sparse learning model, we select the most informative features from the matrix. Specifically, we measure the importance of features using mutual information and introduce an adaptive weight construction strategy to penalize regression coefficients corresponding to each variable adaptively. Moreover, the weighted fusion penalty balances weight differences among correlated variables, reducing the model's overreliance on specific variables and enhancing accuracy. The performance of the proposed method was validated on BCI Competition IV datasets IIa and IIb using the support vector machine. Experimental results demonstrate the effectiveness and superiority of the proposed model compared to the existing models.

Stochastic \({p}\)th Root Approximation of a Stochastic Matrix: A Riemannian Optimization Approach

Stochastic \({p}\)th Root Approximation of a Stochastic Matrix: A Riemannian Optimization Approach

Diffusion time-related structure-function coupling reveals differential association with inter-individual variations in body mass index

Body mass index (BMI) is an indicator of obesity, and recent neuroimaging studies have demonstrated that inter-individual variations in BMI are associated with altered brain structure and function. However, the mechanism underlying the alteration of structure-function correspondence according to BMI is under-investigated. In this study, we studied structural and functional connectivity derived from diffusion MRI tractography and inter-regional correlations of functional MRI time series, respectively. We combined the structural and functional connectivity information using the Riemannian optimization approach. First, the low-dimensional principal eigenvectors (i.e., gradients) of the structural connectivity were generated by applying diffusion map embedding with varying diffusion times. A transformation was identified so that the structural and functional embeddings share the same coordinate system, and subsequently, the functional connectivity matrix was simulated. Then, we generated gradients from the simulated functional connectivity matrix. We found the most apparent cortical hierarchical organization differentiating between low-level sensory and higher-order transmodal regions in the middle of the diffusion time, indicating that the hierarchical organization of the brain may reflect the intermediate mechanisms of mono- and polysynaptic communications. Associations between the functional gradients and BMI were strongest when the hierarchical structure was the most evident. Moreover, the gradient-BMI association map was related to the microstructural features, and the findings indicated that the BMI-related structure-function coupling was significantly associated with brain microstructure, particularly in higher-order transmodal areas. Finally, transcriptomic association analysis revealed the potential biological underpinnings specifying gene enrichment in the striatum, hypothalamus, and cortical cells. Our findings provide evidence that structure-function correspondence is strongly coupled with BMI when hierarchical organization is the most apparent and that the associations are related to the multiscale properties of the brain, leading to an advanced understanding of the neural mechanisms related to BMI.

Multiband Convolutional Riemannian Network with Band-wise Riemannian Triplet Loss for Motor Imagery Classification.

This paper presents a novel motor imagery classification algorithm that uses an overlapping multiscale multiband convolutional Riemannian network with band-wise Riemannian triplet loss to improve classification performance. Despite the superior performance of the Riemannian approach over the common spatial pattern filter approach, deep learning methods that generalize the Riemannian approach have received less attention. The proposed algorithm develops a state-of-the-art multiband Riemannian network that reduces the potential overfitting problem of Riemannian networks, a drawback of Riemannian networks due to their inherent large feature dimension from covariance matrix, by using fewer subbands with discriminative frequency diversity, by inserting convolutional layers before computing the subband covariance matrix, and by regularizing subband networks with Riemannian triplet loss. The proposed method is evaluated using the publicly available datasets, the BCI Competition IV dataset 2a and the OpenBMI dataset. The experimental results confirm that the proposed method improves performance, in particular achieving state-of-the-art classification accuracy among the currently studied Riemannian networks.

LEO- and RIS-Empowered User Tracking: A Riemannian Manifold Approach

LEO- and RIS-Empowered User Tracking: A Riemannian Manifold Approach

A Riemannian Manifold Approach to Constrained Resource Allocation in ISAC

A Riemannian Manifold Approach to Constrained Resource Allocation in ISAC

Brain Decoding over the MEG Signals Using Riemannian Approach and Machine Learning

Brain decoding is an emerging approach for understanding the face perception mechanism in the human brain. Face visual stimuli and perception mechanism are considered as a challenging ongoing research of the neuroscience field. In this study, face/scrambled face visual stimulations were implemented over the sixteen participants to be decoded the face or scrambled face classification using machine learning (ML) algorithms via magnetoencephalography (MEG) signals. This noninvasive and high spatial/temporal resolution signal is a neurophysiological technique which measures the magnetic fields generated by the neuronal activity of the brain. The Riemannian approach was used as a highly promising feature extraction technique. Then Long Short-Term Memory (LSTM), Gated Recurrent Unit (GRU), Convolutional Neural Network (CNN) were employed as deep learning algorithms, Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA) were implemented as shallow algorithms. The improved classification performances are very encouraging, especially for deep learning algorithms. The LSTM and GRU have achieved 92.99% and 91.66% accuracy and 0.977 and 0.973 of the area under the curve (AUC) scores, respectively. Moreover, CNN has yielded 90.62% accuracy. As our best knowledge, the improved outcomes and the usage of the deep learning on the MEG dataset signals from 16 participants are critical to expand the literature of brain decoding after visual stimuli. And this study is the first attempt with these methods in systematic comparison. Moreover, MEG-based Brain-Computer Interface (BCI) approaches may also be implemented for Internet of Things (IoT) applications, including biometric authentication, thanks to the specific stimuli of individual’s brainwaves.

Online Ternary Classification of Covert Speech by Leveraging the Passive Perception of Speech.

Brain-computer interfaces (BCIs) provide communicative alternatives to those without functional speech. Covert speech (CS)-based BCIs enable communication simply by thinking of words and thus have intuitive appeal. However, an elusive barrier to their clinical translation is the collection of voluminous examples of high-quality CS signals, as iteratively rehearsing words for long durations is mentally fatiguing. Research on CS and speech perception (SP) identifies common spatiotemporal patterns in their respective electroencephalographic (EEG) signals, pointing towards shared encoding mechanisms. The goal of this study was to investigate whether a model that leverages the signal similarities between SP and CS can differentiate speech-related EEG signals online. Ten participants completed a dyadic protocol where in each trial, they listened to a randomly selected word and then subsequently mentally rehearsed the word. In the offline sessions, eight words were presented to participants. For the subsequent online sessions, the two most distinct words (most separable in terms of their EEG signals) were chosen to form a ternary classification problem (two words and rest). The model comprised a functional mapping derived from SP and CS signals of the same speech token (features are extracted via a Riemannian approach). An average ternary online accuracy of 75.3% (60% chance level) was achieved across participants, with individual accuracies as high as 93%. Moreover, we observed that the signal-to-noise ratio (SNR) of CS signals was enhanced by perception-covert modeling according to the level of high-frequency ([Formula: see text]-band) correspondence between CS and SP. These findings may lead to less burdensome data collection for training speech BCIs, which could eventually enhance the rate at which the vocabulary can grow.

Revisiting the electronic properties of disclinated graphene sheets

The interplay between topological defects, such as dislocations or disclinations, and the electronic degrees of freedom in graphene have been extensively studied. In the literature, for the study of this kind of problems, it is in general used either a gauge theory or a curved spatial Riemannian geometry approach, where, in the geometric case, the information about the defects is contained in the metric and the spin-connection. However, these topological defects can also be associated with a Riemann–Cartan geometry where curvature and torsion plays an important role. In this article, we study the interplay between a wedge disclination in a planar graphene sheet and the properties of its electronic degrees of freedom. Our approach relies on its relation with elasticity theory through the so called elastic-gauge, where their typical coefficients, as for example the Poisson’s ratio, appear directly in the metric, and consequently also in the electronic spectrum.

Stability in inverse problem of an elastic plate with a curved middle surface

We consider stability in an inverse problem of determining three spatially varying functions including the source term and the mass density for a curved plate by the Riemannian geometrical approach. The stability is derived by the Carleman estimates and observability inequalities. Two kinds of boundary conditions are considered: one is the hinged boundary conditions and the other is the clamped boundary conditions. In particular, the case of the Euler–Bernoulli plate is included.

4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data.

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of objects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered. Also, different deforming objects, hereinafter referred to as 4D surfaces, evolve at different speeds and thus they need to be temporally aligned. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemannian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields (SRNF) where the [Formula: see text] metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces even in the presence of large elastic deformations and large variations in their execution rates; (2) the computation of geodesics between 4D surfaces; (3) the computation of statistical summaries, such as means and modes of variation, of collections of 4D surfaces; and (4) the synthesis of random 4D surfaces. We demonstrate the performance of the proposed framework using 4D facial surfaces and 4D human body shapes.

Functional random effects modeling of brain shape and connectivity

We present a statistical framework that jointly models brain shape and functional connectivity which are two complex aspects of the brain that have been classically studied independently. We adopt a Riemannian modeling approach to account for the non-Euclidean geometry of the space of shapes and the space of connectivity that constrains trajectories of covariation to be valid statistical estimates. In order to disentangle genetic sources of variability from those driven by unique environmental factors, we embed a functional random effects model in the Riemannian framework. We apply the proposed model to the Human Connectome Project dataset to explore spontaneous co-variation between brain shape and connectivity in young healthy individuals.

A singular Riemannian geometry approach to deep neural networks II. Reconstruction of 1-D equivalence classes

We proposed in a previous work a geometric framework to study a deep neural network, seen as sequence of maps between manifolds, employing singular Riemannian geometry. In this paper, we present an application of this framework, proposing a way to build the class of equivalence of an input point: such class is defined as the set of the points on the input manifold mapped to the same output by the neural network. In other words, we build the preimage of a point in the output manifold in the input space. In particular. We focus for simplicity on the case of neural networks maps from n–dimensional real spaces to (n−1)–dimensional real spaces, we propose an algorithm allowing to build the set of points lying on the same class of equivalence. This approach leads to two main applications: the generation of new synthetic data and it may provides some insights on how a classifier can be confused by small perturbation on the input data (e.g. a penguin image classified as an image containing a chihuahua). In addition, for neural networks from 2D to 1D real spaces, we also discuss how to find the preimages of closed intervals of the real line. We also present some numerical experiments with several neural networks trained to perform non-linear regression tasks, including the case of a binary classifier.

A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations

Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely on Euclidean geometry, but in the last years new approaches based on Riemannian geometry have been developed. Motivated by some open problems, we study a particular sequence of maps between manifolds, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the structures induced through pullbacks on the other manifolds of the sequence and on some related quotients. In particular, we show that the pullbacks of the final Riemannian metric to any manifolds of the sequence is a degenerate Riemannian metric inducing a structure of pseudometric space. We prove that the Kolmogorov quotient of this pseudometric space yields a smooth manifold, which is the base space of a particular vertical bundle. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between manifolds implementing neural networks of practical interest and we present some applications of the geometric framework we introduced in the first part of the paper.

Introducing block-Toeplitz covariance matrices to remaster linear discriminant analysis for event-related potential brain–computer interfaces

Objective. Covariance matrices of noisy multichannel electroencephalogram (EEG) time series data provide essential information for the decoding of brain signals using machine learning methods. However, small datasets and high dimensionality make it hard to estimate these matrices. In brain–computer interfaces (BCI) based on event-related potentials (ERP) and a linear discriminant analysis (LDA) classifier, the state of the art covariance estimation uses shrinkage regularization. As this is a general covariance regularization approach, we aim at improving LDA further by better exploiting the domain-specific characteristics of the EEG to regularize the covariance estimates. Approach. We propose to enforce a block-Toeplitz structure for the covariance matrix of the LDA, which implements an assumption of signal stationarity in short time windows for each channel. Main results. An offline re-analysis of data collected from 213 subjects under 13 different event-related potential BCI protocols showed a significantly increased binary classification performance of this ‘ToeplitzLDA’ compared to shrinkage regularized LDA (up to 6 AUC points, p < 0.001) and Riemannian classification approaches (up to 2 AUC points, p < 0.001). In an unsupervised visual speller application, this improvement would translate to a relative reduction of spelling errors by 81% on average for 25 subjects. Additionally, aside from lower memory and reduced time complexity for LDA training, ToeplitzLDA proves to be robust against drastic increases of the number of temporal features. Significance. The proposed covariance estimation allows BCI researchers to improve classification rates and reduce calibration times of BCI protocols using event-related potentials and thus support the usability of corresponding applications. Its lower computational and memory needs could make it a valuable algorithm especially for mobile BCIs.

RNeuMark: A Riemannian EEG Analysis Framework for Neuromarketing

Neuromarketing exploits neuroimaging techniques so as to reinforce the predictive power of conventional marketing tools, like questionnaires and focus groups. Electroencephalography (EEG) is the most commonly encountered neuroimaging technique due to its non-invasiveness, low-cost, and its very recent embedding in wearable devices. The transcription of brainwave patterns to consumer attitude is supported by various signal descriptors, while the quest for profitable novel ways is still an open research question. Here, we suggest the use of sample covariance matrices as alternative descriptors, that encapsulate the coordinated neural activity from distinct brain areas, and the adoption of Riemannian geometry for their handling. We first establish the suitability of Riemannian approach for neuromarketing-related problems and then suggest a relevant decoding scheme for predicting consumers’ choices (e.g., willing to buy or not a specific product). Since the decision-making process involves the concurrent interaction of various cognitive processes and consequently of distinct brain rhythms, the proposed decoder takes the form of an ensemble classifier that builds upon a multi-view perspective, with each view dedicated to a specific frequency band. Adopting a standard machine learning procedure, and using a set of trials (training data) in conjunction with the associated behavior labels (“buy”/ “not buy”), we train a battery of classifiers accordingly. Each classifier is designed to operate in the space recovered from the inter-trial distances of SCMs and to cast a rhythm-depended decision that is eventually combined with the predictions of the rest ones. The demonstration and evaluation of the proposed approach are performed in 2 neuromarketing-related datasets of different nature. The first is employed to showcase the potential of the suggested descriptor, while the second to showcase the decoder’s superiority against popular alternatives in the field.

Convolutional neural network and riemannian geometry hybrid approach for motor imagery classification

The electroencephalogram (EEG) signal is commonly applied in the brain-computer interface (BCI) system of the motor imagery paradigm because it is noninvasive and has a high time resolution. This paper proposes a motor imagery classification method based on convolutional neural networks and Riemannian geometry to overcome the problem of noise and extreme values impacting motor imagery classification performance. The time-domain properties of EEG signals are extracted using multiscale temporal convolutions, whereas the spatial aspects of EEG signals are extracted using multiple convolutional kernels learned by spatial convolution. The extracted features are mapped to a Riemannian manifold space, and bilinear mapping and logarithmic operations are performed on the features to solve the problem of noise and extreme values. The effectiveness of the proposed method is validated using four types of motor imagery in the BCI competition IV dataset 2a to evaluate the classification ability. The experimental results show that the proposed approach has obvious advantages in the classification performance of motor imagery.

A Riemannian approach to predicting brain function from the structural connectome

Ongoing brain function is largely determined by the underlying wiring of the brain, but the specific rules governing this relationship remain unknown. Emerging literature has suggested that functional interactions between brain regions emerge from the structural connections through mono- as well as polysynaptic mechanisms. Here, we propose a novel approach based on diffusion maps and Riemannian optimization to emulate this dynamic mechanism in the form of random walks on the structural connectome and predict functional interactions as a weighted combination of these random walks. Our proposed approach was evaluated in two different cohorts of healthy adults (Human Connectome Project, HCP; Microstructure-Informed Connectomics, MICs). Our approach outperformed existing approaches and showed that performance plateaus approximately around the third random walk. At macroscale, we found that the largest number of walks was required in nodes of the default mode and frontoparietal networks, underscoring an increasing relevance of polysynaptic communication mechanisms in transmodal cortical networks compared to primary and unimodal systems.