Riemannian Geometry Research Articles

A cross-session motor imagery classification method based on Riemannian geometry and deep domain adaptation

Recently, more and more studies have begun to use deep learning to decode and classify EEG signals. The use of deep learning has led to an increase in the classification accuracy of motor imagery (MI), but the problem of taking a long time to calibrate in brain–computer interface (BCI) applications has not been solved. To address this problem, we propose a novel Riemannian geometry and deep domain adaptation network (RGDDANet) for MI classification. Specifically, two one-dimensional convolutions are designed to extract temporal and spatial features from the EEG signals, and then the spatial covariance matrices are utilized to map the extracted features to Riemannian manifolds for processing. In order to align the source and target features’ distributions on the Riemannian manifold, we propose a Symmetric Positive Definite (SPD) matrix mean discrepancy loss (SMMDL) to minimize the distance between two domains. To analyze the feasibility of the method, we conducted extensive experiments on BCIC IV 2a and BCIC IV 2b datasets, respectively, and the results showed that the proposed method achieved better performance than some state-of-the-art methods.

Cross-Sensory EEG Emotion Recognition with Filter Bank Riemannian Feature and Adversarial Domain Adaptation.

Emotion recognition is crucial in understanding human affective states with various applications. Electroencephalography (EEG)-a non-invasive neuroimaging technique that captures brain activity-has gained attention in emotion recognition. However, existing EEG-based emotion recognition systems are limited to specific sensory modalities, hindering their applicability. Our study innovates EEG emotion recognition, offering a comprehensive framework for overcoming sensory-focused limits and cross-sensory challenges. We collected cross-sensory emotion EEG data using multimodal emotion simulations (three sensory modalities: audio/visual/audio-visual with two emotion states: pleasure or unpleasure). The proposed framework-filter bank adversarial domain adaptation Riemann method (FBADR)-leverages filter bank techniques and Riemannian tangent space methods for feature extraction from cross-sensory EEG data. Compared with Riemannian methods, filter bank and adversarial domain adaptation could improve average accuracy by 13.68% and 8.36%, respectively. Comparative analysis of classification results proved that the proposed FBADR framework achieved a state-of-the-art cross-sensory emotion recognition performance and reached an average accuracy of 89.01% ± 5.06%. Moreover, the robustness of the proposed methods could ensure high cross-sensory recognition performance under a signal-to-noise ratio (SNR) ≥ 1 dB. Overall, our study contributes to the EEG-based emotion recognition field by providing a comprehensive framework that overcomes limitations of sensory-oriented approaches and successfully tackles the difficulties of cross-sensory situations.

A geometric approach to the modified Milnor problem

The Milnor Problem (modified) in the theory of group growth asks whether any finite presented group of vanishing algebraic entropy has at most polynomial growth. We show that a positive answer to the Milnor Problem (modified) is equivalent to the Nilpotency conjecture in Riemannian geometry: given [Formula: see text], there exists a constant [Formula: see text] such that if a compact Riemannian [Formula: see text]-manifold [Formula: see text] satisfies that Ricci curvature [Formula: see text], diameter [Formula: see text] and volume entropy [Formula: see text], then the fundamental group [Formula: see text] is virtually nilpotent. We will verify the Nilpotency conjecture in some cases, and we will verify the vanishing gap phenomena for more cases i.e. if [Formula: see text], then [Formula: see text].

Weyl tensors, strongly regular graphs, multiplicative characters, and a quadratic matrix equation

We study solutions of a quadratic matrix equation arising in Riemannian geometry. Let S be a real symmetric n×n-matrix with zeros on the diagonal and let θ be a real number. We construct nonzero solutions (S,θ) of the set of quadratic equations∑kSi,k=0 and ∑kSi,kSk,j+Si,j2=θSi,j for i<j. Our solutions relate the equations to strongly regular graphs, to group rings, and to multiplicative characters of finite fields.

Self-Similar Solutions of a Bianchi Type-III Model with a Perfect Fluid and Cosmic String Cloud in Riemannian Geometry

A Bianchi type-III cosmological model with self-similarity symmetry is investigated in cases with perfect fluid distribution and a cosmic string cloud. We show that this model admits a non-trivial homothetic vector field which possess non-null homothetic bivectors. This homothetic vector field is not parallel to its source vector (current vector). We discussed cases where the homothetic vector is either orthogonal or parallel to the 4-velocity of a fluid element. We solved Einstein’s field equations without making any assumptions on the geometry of space-time, only assuming that it admits self-similarity symmetry. We obtained new exact self-similar solutions for the Bianchi type-III model. We show that all obtained solutions are shear, and space-time is shear free in only the case where the homothetic vector field is parallel to the 4-velocity vector. The kinematical and physical properties of the obtained solutions are discussed.

Relativistic Mechanics in Gravitational Field within Oblate Spheroidal Coordinates Based upon Riemannian Geometry for Rotating Homogeneous Mass Distribution

The emergence of the geometrical theory of gravitation (general relativity) by Albert Einstein in his quest to unite special relativity and the Newtonian law of universal gravitation has led to several Mathematical approaches for the exact and analytical solution for all gravitational fields in nature. The first and the most famous analytical solution was the Schwarzchild’s which can be constructed by finding a mapping where the metric tensor takes a simple form i.e. the vanishing of the non-diagonal elements. In this paper, we construct exact solution of the Einstein geometrical gravitational field equation using Riemannian metric tensor called the golden metric tensor that was first developed by Howusu, in the year 2009, for the rotating homogeneous mass distribution within oblate Spheroidal Coordinates. The equations of motion for test particles in the Oblate Spheroidal Geometry were derived using the coefficient of affine connection. Then the law of conservation of momentum and energy are equivalently formulated using the generalized Lagrangian as compared to the analytical solution of the Schwarzchild’s gravitational field. We also derived the planetary equation of motion in the equatorial plane of the Oblate Spheroidal body for this gravitational field.

Conformal invariance and phenomenology of particle creation: Weyl geometry vs. Riemannian geometry

Conformal invariance and phenomenology of particle creation: Weyl geometry vs. Riemannian geometry

Конформная инвариантность и феноменология рождения частиц: сравнение геометрии Вейля с римановой геометрией

На примере действия идеальной жидкости с переменным числом частиц изучено феноменологическое описание процессов рождения частиц на фоне сильных внешних полей, в частности гравитационного и скалярного полей. Эта модель рассматривается как для геометрии Вейля, так и для римановой геометрии. Новый инвариант, связанный с взаимодействием вектора Вейля с частицами, инкорпорирован в действие материи. Показана конформная инвариантность слагаемого в действии материи, ответственного за рождение частиц.

General teleparallel gravity from Finsler geometry

Riemannian geometry, as a basis for general relativity, can be obtained from the more general Finsler geometry in terms of the Cartan connection and Chern connection, as discussed frequently in the literature. However, there are other gravity theories that can be made to be equivalent to general relativity but are based on non-Riemannian geometry. Famous examples are the Teleparallel and Symmetric Teleparallel gravity theories. In this paper, we show how to obtain the geometry for Teleparallel gravity from Finsler geometry in terms of a ‘Teleparallel type’ connection.

Cure-GNN: A Robust Curvature-Enhanced Graph Neural Network Against Adversarial Attacks

Graph neural networks (GNNs) are a specialized type of deep learning models on graphs by learning aggregations over neighbor nodes. However, recent studies reveal that the performance of GNNs are severely deteriorated by injecting adversarial examples. Hence, improving the robustness of GNNs is of significant importance. Prior works are devoted to reducing the influence of direct adversaries which are adversarial attacks by positioning a node's one-hop neighbors, yet these approaches are limited in protecting GNNs from indirect adversarial attacks within a node's multi-hop neighbors. In this work, we approach this problem from a new angle by exploring the graph Ricci curvature, which can characterize the relationships of both direct and indirect links from any two nodes' neighborhoods in the Riemannian space. We first investigate the distinguishable properties of adversarial attacks with graph Ricci curvature distribution. Then, a novel defense framework called Cure-GNN is proposed to detect and mitigate adversarial effects. Cure-GNN discerns the distinction between adversarial edges and normal edges via computing curvature, and merges it into the node features reconstructed by a residual learning framework. Extensive experiments over real-world datasets on node classification task demonstrate the efficacy of Cure-GNN and achieves superiority to the state-of-the-arts without incurring high complexity.

Huygens’ equations and the gradient-flow equations in information geometry

In this paper, we revisit the relation between the gradient-flow equations and Hamilton’s equations in information geometry. By regarding the gradient-flow equations as Huygens’ equations in geometric optics, we have related the gradient flows to the geodesic flows induced by the geodesic Hamiltonian in an appropriate Riemannian geometry. The original evolution parameter [Formula: see text] in the gradient-flow equations is related to the arc-length parameter in the associated Riemannian manifold by Jacobi–Maupertuis transformation. As a by-product, the relation between the gradient-flow equation and replicator equations is found.

Dimension selection for EEG classification in the SPD Riemannian space based on PSO

Several classification methods in the task of electroencephalogram (EEG) classification represent input features as symmetric positive definite (SPD) matrices. By translating the data into a Riemannian space, the classification methods have obtained superior performance; however, the high dimensionality of the data typically results in low computational efficiency and degraded classification performance. Therefore, this paper proposes a novel dimension selection method named dimension selection for SPD matrices on Riemannian manifold (DSSR). DSSR can eliminate redundant dimensions for SPD matrices in the Riemannian space based on the framework of particle swarm optimization (PSO). First, we treat one column of the SPD matrix and its corresponding row as a group, where upon the PSO is used to find optimal groups. Next, we use Riemannian classifiers to evaluate the goodness of the selected groups. Finally, the dimension selected SPD matrices are classified by the generalized learning Riemannian space quantization (GLRSQ). Experimental results on two motor imagery datasets show the superior performance of the proposed methods over EEG classification. Neuroscience experiments prove that the optimal dimensions selected by our proposed method are consistent with the findings of brain science research.

Explicit harmonic morphisms and p-harmonic functions from the complex and quaternionic Grassmannians

We construct explicit complex-valued p-harmonic functions and harmonic morphisms on the classical compact symmetric complex and quaternionic Grassmannians. The ingredients for our construction method are joint eigenfunctions of the classical Laplace–Beltrami and the so-called conformality operator. A known duality principle implies that these p-harmonic functions and harmonic morphisms also induce such solutions on the Riemannian symmetric non-compact dual spaces.

Research on point cloud simplification algorithm for ring forgings based on joint entropy evaluation

There are numerous redundant points in the point cloud model of ring forgings obtained by 3D laser scanner. How to remove the redundant points while keeping the model characteristics unchanged is a critical issue. This paper proposes a point cloud simplification algorithm based on the joint entropy evaluation theory. Firstly, the K-D tree is used to search for the K-neighbors of the sampled points. Secondly, a surface is fitted to the spatial neighborhood of the sampled points using the least squares method. The curvature operator of the sampled points is derived on the fitted surface using Riemannian geometry theory. After that, an energy operator is defined by using the normal vectors and distances of the sampled points and their neighborhood points. The joint entropy values of all points in the model are determined based on the probability distributions of these two operators in the local neighborhood. Finally, the data points are sorted by entropy value. Data points with high entropy values are put into the data set U1. Data points with low entropy values are clustered through the K-Means algorithm of swarm optimization. The redundant points outside the cluster centers are removed, and the cluster centers are put into the data set U2. The final simplification results are obtained by integrating data sets U1 and U2. The experimental results show that the point cloud simplification algorithm proposed in this paper is effective and feasible.

Multimodal Approach for Pilot Mental State Detection Based on EEG.

The safety of flight operations depends on the cognitive abilities of pilots. In recent years, there has been growing concern about potential accidents caused by the declining mental states of pilots. We have developed a novel multimodal approach for mental state detection in pilots using electroencephalography (EEG) signals. Our approach includes an advanced automated preprocessing pipeline to remove artefacts from the EEG data, a feature extraction method based on Riemannian geometry analysis of the cleaned EEG data, and a hybrid ensemble learning technique that combines the results of several machine learning classifiers. The proposed approach provides improved accuracy compared to existing methods, achieving an accuracy of 86% when tested on cleaned EEG data. The EEG dataset was collected from 18 pilots who participated in flight experiments and publicly released at NASA's open portal. This study presents a reliable and efficient solution for detecting mental states in pilots and highlights the potential of EEG signals and ensemble learning algorithms in developing cognitive cockpit systems. The use of an automated preprocessing pipeline, feature extraction method based on Riemannian geometry analysis, and hybrid ensemble learning technique set this work apart from previous efforts in the field and demonstrates the innovative nature of the proposed approach.

Differential Geometry and Its Application

We have launched a Special Issue of Axioms which focuses on the generalization of Riemannian spaces and their mappings [...]

Spatiotemporal analysis using Riemannian composition of diffusion operators

Multivariate time-series have become abundant in recent years, as many data-acquisition systems record information through multiple sensors simultaneously. In this paper, we assume the variables pertain to some geometry and present an operator-based approach for spatiotemporal analysis. Our approach combines three components that are often considered separately: (i) manifold learning for building operators representing the geometry of the variables, (ii) Riemannian geometry of symmetric positive-definite matrices for multiscale composition of operators corresponding to different time samples, and (iii) spectral analysis of the composite operators for extracting different dynamic modes. We propose a method that is analogous to the classical wavelet analysis, which we term Riemannian multi-resolution analysis (RMRA). We provide some theoretical results on the spectral analysis of the composite operators, and we demonstrate the proposed method on simulations and on real data.

How to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Kraus-operator geometry of SL(2,C)

The generalized Q-function of a spin system can be considered the outcome probability distribution of a state subjected to a measurement represented by the spin-coherent-state (SCS) positive-operator-valued measure (POVM). As fundamental as the SCS POVM is to the 2-sphere phase-space representation of spin systems, it has only recently been reported that the SCS POVM can be performed for any spin system by continuous isotropic measurement of the three total spin components [E. Shojaee, C. S. Jackson, C. A. Riofrio, A. Kalev, and I. H. Deutsch, Phys. Rev. Lett. 121, 130404 (2018)]. This article develops the theoretical details of the continuous isotropic measurement and places it within the general context of curved-phase-space correspondences for quantum systems. The analysis is in terms of the Kraus operators that develop over the course of a continuous isotropic measurement. The Kraus operators of any spin j are shown to represent elements of the Lie group SL(2,C)&amp;#x2245;Spin(3,C), a complex version of the usual unitary operators that represent elements of SU(2)&amp;#x2245;Spin(3,R). Consequently, the associated POVM elements represent points in the symmetric space SU(2)&amp;#x2216;SL(2,C), which can be recognized as the 3-hyperboloid. Three equivalent stochastic techniques, (Wiener) path integral, (Fokker-Planck) diffusion equation, and stochastic differential equations, are applied to show that the continuous isotropic POVM quickly limits to the SCS POVM, placing spherical phase space at the boundary of the fundamental Lie group SL(2,C) in an operationally meaningful way. Two basic mathematical tools are used to analyze the evolving Kraus operators, the Maurer-Cartan form, modified for stochastic applications, and the Cartan, decomposition associated with the symmetric pair SU(2) ⊂ SL(2,C). Informed by these tools, the three schochastic techniques are applied directly to the Kraus operators in a representation-independent – and therefore geometric – way (independent of any spectral information about the spin components).The Kraus-operator-centric, geometric treatment applies not just to SU(2) ⊂ SL(2,C), but also to any compact semisimple Lie group and its complexification. The POVM associated with the continuous isotropic measurement of Lie-group generators thus corresponds to a type-IV globally Riemannian symmetric space and limits to the POVM of generalized coherent states. This generalization is the focus of a sequel to this article.

Riemannian laplacian in rotational circular cylindrical coordinates for all gravitational fields in nature based upon the great metric tensor

Most mathematicians’ and physicists’ formulations are based upon Euclidean Geometry for applications in various fields. In this paper, I formulate a new Laplacian operator in Rotational Circular Cylindrical Coordinates which is built upon the Riemannian Geometry, based upon the Great metric tensor, which is more mathematically complete and unique in all gravitational fields for applications in mechanics.

Unique continuation inequalities for Schrödinger equation on Riemannian symmetric spaces of noncompact type

Unique continuation inequalities for Schrödinger equation on Riemannian symmetric spaces of noncompact type