Riemannian Geometric 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.

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.

A Riemannian Geometric Approach to Blind Signal Recovery for Grant-Free Radio Network Access

We propose a new nonconvex framework for blind multiple signal demixing and recovery. The proposed Riemannian geometric approach extends the well known constant modulus algorithm to facilitate grant-free wireless access for simultaneous demixing and recovery of multiple signal demixing and recovery. We formulate the problem as non-convex problem optimization problem integrated with the signal orthogonality constraint in the form of Riemannian Orthogonal CMA (ROCMA). Unlike traditional stochastic gradient solutions that require large data samples, parameter tuning, and careful initialization, we leverage Riemannian geometry and transform the orthogonality requirement of recovered signals into a Riemannian manifold optimization. Our solution demonstrates full recovery of multiple access signals without large data sample size or special initialization with high probability of success.

Riemannian geometric approach to optimal binocular rotation, pyramid based interpolation and bio-mimetic pan-tilt movement

Over the past several years, we have been studying the problem of optimally rotating a rigid sphere about its center, where the rotation is actuated by a triplet of external torques acting on the body. The control objective is to repeatedly direct a suitable radial vector, called the gaze vector, towards a stationary point target in IR3. The orientation of the sphere is constrained to lie in a suitable submanifold of SO(3). Historically, the constrained rotational movements were studied by physiologists in the nineteenth century, interested in eye and head movements. In this paper we revisit the gaze control problem, where two visual sensors, are tasked to simultaneously stare at a point target in the visual space. The target position changes discretely and the problem we consider is how to reorient the gaze directions of the sensors, along the optimal pathway of the human eyes, to the new location of the target. This is done by first solving an optimal control problem on the human binocular system. Next, we use these optimal control and show that a pan-tilt system can be controlled to follow the gaze trajectory of the human eye requiring a nonlinear static feedback of the pan and tilt angles and their derivatives. Our problem formulation uses a new Riemannian geometric description of the orientation space. The paper also introduces a new, pyramid based interpolation method, to implement the optimal controller.

Target Detection Through Riemannian Geometric Approach With Application to Drone Detection

Radar detection of small drones in presence of noise and clutter is considered from a differential geometry viewpoint. The drone detection problem is challenging due to low radar cross section (RCS) of drones, especially in cluttered environments and when drones fly low and slow in urban areas. This paper proposes two detection techniques, the Riemannian-Brauer matrix (RBM) and the angle-based hybrid-Brauer (ABHB), to improve the probability of drone detection under small sample size and low signal-to-clutter ratio (SCR). These techniques are based on the regularized Burg algorithm (RBA), the Brauer disc (BD) theorem, and the Riemannian mean and distance. Both techniques exploit the RBA to obtain a Toeplitz Hermitian positive definite (THPD) covariance matrix from each snapshot and apply the BD theorem to cluster the clutter-plus-noise THPD covariance matrices. The proposed Riemannian-Brauer matrix technique is based on the Riemannian distance between the Riemannian mean of clutter-plus-noise cluster and potential targets. The proposed angle-based hybrid-Brauer technique uses the Euclidean tangent space and the Riemannian geodesical distances between the Riemannian mean, the Riemannian median and the potential target point. The angle at the potential target on the manifold is computed using the law of cosines on the manifold. The proposed detection techniques show advantage over the fast Fourier transform, the Riemannian distance-based matrix and the Kullback-Leibler (KLB) divergence detectors. The validity of both proposed techniques are demonstrated with real data.

A Riemannian Geometric Approach to Output Tracking for Nonholonomic Systems

The problem of designing coordinate-invariant output tracking control laws for nonholonomic mechanical systems is addressed. The velocity constrained Euler-Lagrange equations of motion are expressed through a constrained affine connection which is compatible with the kinetic energy Riemannian metric. This formalism is used in designing an output tracking control law via backstepping, which is shown to guarantee exponential stability when the initial distance between the output and reference trajectory is within injectivity radius of the output manifold. In particular this enables almost-global tracking when the output manifold is a rank-1 symmetric space. The control law is intrinsic to the Riemannian structure, and is explicitly constructed. The control law is applied to the problem of tracking the reduced attitude of a rigid body with a nonholonomic velocity constraint. Numerical simulations illustrating the tracking performance are presented.

Radiologic image-based statistical shape analysis of brain tumours.

We propose a curve-based Riemannian geometric approach for general shape-based statistical analyses of tumours obtained from radiologic images. A key component of the framework is a suitable metric that enables comparisons of tumour shapes, provides tools for computing descriptive statistics and implementing principal component analysis on the space of tumour shapes and allows for a rich class of continuous deformations of a tumour shape. The utility of the framework is illustrated through specific statistical tasks on a data set of radiologic images of patients diagnosed with glioblastoma multiforme, a malignant brain tumour with poor prognosis. In particular, our analysis discovers two patient clusters with very different survival, subtype and genomic characteristics. Furthermore, it is demonstrated that adding tumour shape information to survival models containing clinical and genomic variables results in a significant increase in predictive power.

Stabilization of the transmission wave/plate equation with variable coefficients

We consider the stabilization of the transmission wave/plate equation with variable coefficients and a general transmission boundary. By virtue of the Riemannian geometric approach, we obtain some stability results for nonlinear feedbacks acting on the regions under suitable geometric conditions.

Thermodynamic fluctuations as a probe for the study of gravitational galaxy clustering

We derive the physical form of the clustering parameter on rigorous mathematical grounds from two approaches. One result is a direct consequence of the solution of a differential equation and the other follows from the calculation of thermodynamic metric elements using Riemannian Geometric approach. Both these approaches lead to the same physical form of the clustering parameter as hypothesized in the Gravitational Quasi-Equilibrium Distribution (GQED) theory. These results lend a strong support to GQED theory and its successor: the statistical mechanical description of the cosmological many-body problem.

Publisher's Note: Erratum: Riemannian geometric approach to human arm dynamics, movement optimization, and invariance [Phys. Rev. E83, 031927 (2011)] [Phys. Rev. E85, 019907 (2012)

Received 15 February 2012DOI:https://doi.org/10.1103/PhysRevE.85.029904©2012 American Physical Society

Erratum: Riemannian geometric approach to human arm dynamics, movement optimization, and invariance [Phys. Rev. E83, 031927 (2011)

Erratum: Riemannian geometric approach to human arm dynamics, movement optimization, and invariance [Phys. Rev. E<b>83</b>, 031927 (2011)

Riemannian geometric approach to human arm dynamics, movement optimization, and invariance

We present a generally covariant formulation of human arm dynamics and optimization principles in Riemannian configuration space. We extend the one-parameter family of mean-squared-derivative (MSD) cost functionals from Euclidean to Riemannian space, and we show that they are mathematically identical to the corresponding dynamic costs when formulated in a Riemannian space equipped with the kinetic energy metric. In particular, we derive the equivalence of the minimum-jerk and minimum-torque change models in this metric space. Solutions of the one-parameter family of MSD variational problems in Riemannian space are given by (reparameterized) geodesic paths, which correspond to movements with least muscular effort. Finally, movement invariants are derived from symmetries of the Riemannian manifold. We argue that the geometrical structure imposed on the arm's configuration space may provide insights into the emerging properties of the movements generated by the motor system.

Fluctuations of non-point mass galaxy distribution

We examine cosmic energy equation for extended galaxy structures on the basis of different models of universe. We also extend the power spectrum and density fluctuations for extended structure by introducing softening parameter both for linear and non-linear regimes. The results are compared with earlier results of point mass structures. It is found that softening parameters introduced in the theory influence the thermodynamic fluctuation theory. Results obtained with spectrum analysis are also compared with Riemannian geometric approach (Ruppeiner in Rev. Mod. Phys. 67:605, 1995) to the galaxy clustering. The singular solutions of thermodynamic fluctuation results can be interpreted on the basis of power spectrum analysis in terms of power index law of two point correlation function.

A Riemannian-Geometric Approach for Intelligent Control and Fingertip Design of Multi-fingered Hands

Based upon the analysis of modeling and control of two-dimensional (2-D) grasping and manipulation of arbitrary rigid objects under rolling contact constraints, geometrical conditions for the design of desired fingertip shapes of robot fingers are discussed from the Riemannian-geometric standpoint. A required condition is given as an inequality expressed in terms of quantities of the second fundamental form of fingertip contour curves, although the quantities do not enter into the Euler–Lagrange equation of motion of the overall fingers/object system. Satisfaction of the inequality is necessary for stabilization of grasping by using fingers–thumb opposable control signals without use of external sensings or knowledge of an object to be grasped. At the same time, asymptotic convergence of a solution to the closed-loop dynamics of 2-D precision prehension by a pair of multi-joint robot fingers is proved under rolling contact constraints and the existence of redundancy in the system's degrees of freedom. This is regarded as an extension of the Dirichlet–Lagrange stability theorem to a dynamical system with redundancy and geometric constraints.

Riemannian geometric approach to chaos in SU(2) Yang-Mills theory

Based on the Riemannian geometric approach to Hamiltonian systems with many degrees of freedom, we study a chaotic nature of the SU(2) Yang-Mills field. Particularly, we study the Lyapunov exponent of the Wu-Yang magnetic-monopole solution of the SU(2) Yang-Mills field equation by use of an analytic formula which is determined by the average Ricci curvature and its fluctuation on the Riemannian manifold. It is shown that the system is chaotic from the positive values of the Lyapunov exponent. Furthermore, we find that the energy dependence of Lyapunov exponents exhibits a crossover phenomenon. By using the linear stability analysis, we point out that this crossover is related to the instability of the monopole solution.

Jacobi fields on statistical manifolds of negative curvature

Two entropic dynamical models are considered. The geometric structure of the statistical manifolds underlying these models is studied. It is found that in both cases, the resulting metric manifolds are negatively curved. Moreover, the geodesics on each manifold are described by hyperbolic trajectories. A detailed analysis based on the Jacobi equation for geodesic spread is used to show that the hyperbolicity of the manifolds leads to chaotic exponential instability. A comparison between the two models leads to a relation among statistical curvature, stability of geodesics and relative entropy-like quantities. Finally, the Jacobi vector field intensity and the entropy-like quantity are suggested as possible indicators of chaoticity in the ED models due to their similarity to the conventional chaos indicators based on the Riemannian geometric approach and the Zurek–Paz criterion of linear entropy growth, respectively.

Geometric Control of Mechanical Systems

This talk will outline a comprehensive set of modeling, analysis and design techniques for a class of mechanical systems. We concern ourselves with simple mechanical control systems, that is, systems whose Lagrangian is kinetic energy minus potential energy. Example devices include robotic manipulators, aerospace and underwater vehicles, and mechanisms that locomote exploiting nonholonomic constraints. Borrowing techniques from nonlinear control and geometric mechanics, we propose a coordinateinvariant control theory for this class of systems. First, we take a Riemannian geometric approach to modeling systems dened on smooth manifolds, subject to nonholonomic constraints, external forces and control forces. We also model mechanical systems on groups and symmetries. Second, we analyze some control-theoretic properties of this class of systems, including controllability, averaged response to oscillatory controls, and kinematic reductions. Finally, we exploit the modeling and analysis results to tackle control design problems. Starting from controllability and kinematic reduction assumptions we propose some algorithms for generating and tracking trajectories.

Indicator of chaos based on the Riemannian geometric approach

Using the Riemannian geometric approach to Hamiltonian systems, we show that the empirical indicator of chaos proposed by Kosloff and Rice [J. Chem. Phys. 74, 1947 (1981)], an improved version of the well-known Toda-Brumer criterion, is equivalent to sectional curvature of the Jacobi equation when the Eisenhart metric is chosen for the Riemannian manifold. Further, we present a relation among the sectional curvature, the Lyapunov exponents, and the Kolmogorov-Sinai entropy. By using this relation, the empirical indicator by Kosloff and Rice, which is local, can be used as a global indicator of chaos.

Dual scaling of comparison and reference stimuli in multi-dimensional psychological space

Dzhafarov and Colonius (Psychol. Bull. Rev. 6 (1999)239; J. Math. Psychol. 45(2001)670) proposed a theory of Fechnerian scaling of the stimulus space based on the psychometric (discrimination probability) function of a human subject in a same–different comparison task. Here, we investigate a related but different paradigm, namely, referent–probe comparison task, in which the pair of stimuli ( x and y) under comparison assumes substantively different psychological status, one serving as a referent and the other as a probe. The duality between a pair of psychometric functions, arising from assigning either x or y to be the fixed reference stimulus and the other to be the varying comparison stimulus, and the 1-to-1 mapping between the two stimulus spaces X and Y under either assignment are analyzed. Following Dzhafarov and Colonius, we investigate two properties characteristic of a referent–probe comparison task, namely, (i) Regular cross-minimality—for the pair of stimulus values involved in referent–probe comparison, each minimizes a discrimination probability function where the other is treated as the fixed reference stimulus; (ii) Nonconstant self-similarity—the value of the discrimination probability function at its minima is a nonconstant function of the reference stimulus value. For the particular form of psychometric functions investigated, it is shown that imposing the condition of regular cross-minimality on the pair of psychometric functions forces a consistent (but otherwise still arbitrary) mapping between X and Y, such that it is independent of the assignment of reference/comparison status to x and to y. The resulting psychometric differentials under both assignments are equal, and take an asymmetric, dualistic form reminiscent of the so-called divergence measure that appeared in the context of differential geometry of the probability manifold with dually flat connections (Differential Geometric Methods in Statistics, Lecture Notes in Statistics, Vol. 28, Springer, New York, 1985). The pair of divergence functions on X and on Y, respectively, induce a Riemannian metric in the small, with psychometric order (defined in Dzhafarov & Colonius, 1999) equal to 2. The difference between the Finsler–Riemann geometric approach to the stimulus space (Dzhafarov & Colonius, 1999) and this dually-affine Riemannian geometric approach to the dual scaling of the comparison and the reference stimuli is discussed.

Chaos based on Riemannian geometric approach to Abelian-Higgs dynamical system.

Based on the Riemannian geometric approach, we study chaos of the Abelian-Higgs dynamical system derived from a classical field equation consisting of a spatially homogeneous Abelian gauge field and Higgs field. Using the global indicator of chaos formulated by the sectional curvature of the ambient manifold, we show that this approach brings the same qualitative and quantitative information about order and chaos as has been provided by the Lyapunov exponents in the conventional and phenomenological approach. We confirm that the mechanism of chaos is a parametric instability of the system. By analyzing a close relation between the sectional curvature and the Gaussian curvature, we point out that the Toda-Brumer criterion becomes a sufficient condition to the criterion based on this geometric approach as to the stability condition.