Tangent Space Research Articles

Wind Finslerian Structures: From Zermelo’s Navigation to the Causality of Spacetimes

The notion of wind Finslerian structure Σ \Sigma is developed; this is a generalization of Finsler metrics (and Kropina ones) where the indicatrices at the tangent spaces may not contain the zero vector. In the particular case that these indicatrices are ellipsoids, called here wind Riemannian structures, they admit a double interpretation which provides: (a) a model for classical Zermelo’s navigation problem even when the trajectories of the moving objects (planes, ships) are influenced by strong winds or streams, and (b) a natural description of the causal structure of relativistic spacetimes endowed with a non-vanishing Killing vector field K K (SSTK splittings), in terms of Finslerian elements. These elements can be regarded as conformally invariant Killing initial data on a partial Cauchy hypersurface. The links between both interpretations as well as the possibility to improve the results on one of them using the other viewpoint are stressed. The wind Finslerian structure Σ \Sigma is described in terms of two (conic, pseudo) Finsler metrics, F F and F l F_l , the former with a convex indicatrix and the latter with a concave one. Notions such as balls and geodesics are extended to Σ \Sigma . Among the applications, we obtain the solution of Zermelo’s navigation with arbitrary time-independent wind, metric-type properties for Σ \Sigma (distance-type arrival function, completeness, existence of minimizing, maximizing or closed geodesics), as well as description of spacetime elements (Cauchy developments, black hole horizons) in terms of Finslerian elements in Killing initial data. A general Fermat’s principle of independent interest for arbitrary spacetimes, as well as its applications to SSTK\xspace spacetimes and Zermelo’s navigation, are also provided.

PCDNF: Revisiting Learning-Based Point Cloud Denoising via Joint Normal Filtering.

Point cloud denoising is a fundamental and challenging problem in geometry processing. Existing methods typically involve direct denoising of noisy input or filtering raw normals followed by point position updates. Recognizing the crucial relationship between point cloud denoising and normal filtering, we re-examine this problem from a multitask perspective and propose an end-to-end network called PCDNF for joint normal filtering-based point cloud denoising. We introduce an auxiliary normal filtering task to enhance the network's ability to remove noise while preserving geometric features more accurately. Our network incorporates two novel modules. First, we design a shape-aware selector to improve noise removal performance by constructing latent tangent space representations for specific points, taking into account learned point and normal features as well as geometric priors. Second, we develop a feature refinement module to fuse point and normal features, capitalizing on the strengths of point features in describing geometric details and normal features in representing geometric structures, such as sharp edges and corners. This combination overcomes the limitations of each feature type and better recovers geometric information. Extensive evaluations, comparisons, and ablation studies demonstrate that the proposed method outperforms state-of-the-art approaches in both point cloud denoising and normal filtering.

EEG Motor Imagery Classification: Tangent Space with Gate-Generated Weight Classifier.

Individuals grappling with severe central nervous system injuries often face significant challenges related to sensorimotor function and communication abilities. In response, brain-computer interface (BCI) technology has emerged as a promising solution by offering innovative interaction methods and intelligent rehabilitation training. By leveraging electroencephalographic (EEG) signals, BCIs unlock intriguing possibilities in patient care and neurological rehabilitation. Recent research has utilized covariance matrices as signal descriptors. In this study, we introduce two methodologies for covariance matrix analysis: multiple tangent space projections (M-TSPs) and Cholesky decomposition. Both approaches incorporate a classifier that integrates linear and nonlinear features, resulting in a significant enhancement in classification accuracy, as evidenced by meticulous experimental evaluations. The M-TSP method demonstrates superior performance with an average accuracy improvement of 6.79% over Cholesky decomposition. Additionally, a gender-based analysis reveals a preference for men in the obtained results, with an average improvement of 9.16% over women. These findings underscore the potential of our methodologies to improve BCI performance and highlight gender-specific performance differences to be examined further in our future studies.

Transmit beampattern design for FDA-MIMO radar using complex manifold optimization

Frequency diverse array multiple-input multiple-output (FDA-MIMO) radar can produce time-stationary range-angle dependent beampattern. The problem with these beampatterns is the repeating lobes in the angle-range domain. Different methods are proposed to obtain better beampattern in terms of grating lobe and sidelobe suppression by using frequency offsets. However, these methods mostly choose the frequency offsets using an analytic expression and do not offer a control on the resulting range beampattern. In this paper, a new method for transmit beampattern design in range is presented. This method transfers the design problem into a tangential space and solves the problem using complex manifolds through an optimization procedure. The proposed approach considers amplitude tapering in the transmitter antenna elements and the practical instrumented range of the radar. Range beampattern constraints can be defined conveniently leading to an effective design process. Several simulations are done to show that the proposed method can significantly improve the beamwidth and sidelobe level in range patterns.

Riemannian Geodesic Discriminant Analysis–Minimum Riemannian Mean Distance: A Robust and Effective Method Leveraging a Symmetric Positive Definite Manifold and Discriminant Algorithm for Image Set Classification

This study introduces a novel method for classifying sets of images, called Riemannian geodesic discriminant analysis–minimum Riemannian mean distance (RGDA-MRMD). This method first converts image data into symmetric positive definite (SPD) matrices, which capture important features related to the variability within the data. These SPD matrices are then mapped onto simpler, flat spaces (tangent spaces) using a mathematical tool called the logarithm operator, which helps to reduce their complexity and dimensionality. Subsequently, regularized local Fisher discriminant analysis (RLFDA) is employed to refine these simplified data points on the tangent plane, focusing on local data structures to optimize the distances between the points and prevent overfitting. The optimized points are then transformed back into a complex, curved space (SPD manifold) using the exponential operator to enhance robustness. Finally, classification is performed using the minimum Riemannian mean distance (MRMD) algorithm, which assigns each data point to the class with the closest mean in the Riemannian space. Through experiments on the ETH-80 (Eidgenössische Technische Hochschule Zürich-80 object category), AFEW (acted facial expressions in the wild), and FPHA (first-person hand action) datasets, the proposed method demonstrates superior performance, with accuracy scores of 97.50%, 37.27%, and 88.47%, respectively. It outperforms all the comparison methods, effectively preserving the unique topological structure of the SPD matrices and significantly boosting image set classification accuracy.

Dynamic community detection algorithm based on hyperbolic graph convolution

PurposeCommunity detection is a key factor in analyzing the structural features of complex networks. However, traditional dynamic community detection methods often fail to effectively solve the problems of deep network information loss and computational complexity in hyperbolic space. To address this challenge, a hyperbolic space-based dynamic graph neural network community detection model (HSDCDM) is proposed.Design/methodology/approachHSDCDM first projects the node features into the hyperbolic space and then utilizes the hyperbolic graph convolution module on the Poincaré and Lorentz models to realize feature fusion and information transfer. In addition, the parallel optimized temporal memory module ensures fast and accurate capture of time domain information over extended periods. Finally, the community clustering module divides the community structure by combining the node characteristics of the space domain and the time domain. To evaluate the performance of HSDCDM, experiments are conducted on both artificial and real datasets.FindingsExperimental results on complex networks demonstrate that HSDCDM significantly enhances the quality of community detection in hierarchical networks. It shows an average improvement of 7.29% in NMI and a 9.07% increase in ARI across datasets compared to traditional methods. For complex networks with non-Euclidean geometric structures, the HSDCDM model incorporating hyperbolic geometry can better handle the discontinuity of the metric space, provides a more compact embedding that preserves the data structure, and offers advantages over methods based on Euclidean geometry methods.Originality/valueThis model aggregates the potential information of nodes in space through manifold-preserving distribution mapping and hyperbolic graph topology modules. Moreover, it optimizes the Simple Recurrent Unit (SRU) on the hyperbolic space Lorentz model to effectively extract time series data in hyperbolic space, thereby enhancing computing efficiency by eliminating the reliance on tangent space.

Framed DDF operators and the general solution to Virasoro constraints

We define the framed DDF operators by introducing the concept of local frames in the usual formulation of DDF operators. In doing so it is possible to completely decouple the DDF operators from the associated tachyon and show that they are good zero-dimensional conformal operators. These framed DDFs allow for an explicit covariant formulation of the general solution of the Virasoro constraints both on-shell and off-shell which depends on a unique set of tangent space lightcone polarizations. This is possible since the frame allows us to embed the SO(D-2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SO(D-2)$$\\end{document} lightcone physical polarizations into the SO(1,D-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SO(1,D-1)$$\\end{document} covariant ones in the most general way. The solution to the Virasoro constraints is not in the gauge that is usually used since the states obtained from DDF operators are generically the sum of terms which are partially transverse due to the presence of a projector but not traceless and terms which are partially traceless but not transverse. We then show that different embeddings are connected by states generated by improved Brower operators both on shell and off shell. Brower states are null on-shell and equivalent to BRST exact states. Therefore on shell one embedding is sufficient to compute all amplitudes except the ones with vanishing momentum. Off-shell a change of frame cannot be obtained by null states but only by Brower states. Improved Brower operators generate a gauge invariance associated with local frame rotations, which replaces the usual gauge invariance generated by Virasoro operators or BRST. To check the identification, we verify the matching of the expectation value of the second Casimir of the Poincaré group for some lightcone states with the corresponding covariant states built using the framed DDFs.

Quantifying macrostructures in viscoelastic sub-diffusive flows

We present a theory to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia) for viscoelastic sub-diffusive flows, by introducing a mathematically consistent decomposition of the polymer conformation tensor, into the so-called structure tensor. Our approach bypasses an inherent problem in the standard arithmetic decomposition, namely, the fluctuating conformation tensor fields may not be positive definite and hence, do not retain their physical meaning. Using well-established results in matrix analysis, the space of positive definite matrices is transformed into a Riemannian manifold by defining and constructing a geodesic via the inner product on its tangent space. This geodesic is utilized to define three scalar invariants of the structure tensor, which do not suffer from the caveats of the regular invariants (such as trace and determinant) of the polymer conformation tensor. First, we consider the problem of formulating perturbative expansions of the structure tensor using the geodesic, which is consistent with the Riemannian manifold geometry. A constraint on the maximum time, during which the evolution of the perturbative solution can be well approximated by linear theory along the Euclidean manifold, is found. A comparison between the linear and the nonlinear dynamics, identifies the role of nonlinearities in initiating the symmetry breaking of the flow variables about the centerline. Finally, fully nonlinear simulations of the viscoelastic sub-diffusive channel flows, underscore the advantage of using these invariants in effectively quantifying the macrostructures.

Robust Hashing With Local Tangent Space Alignment for Image Copy Detection

Robust Hashing With Local Tangent Space Alignment for Image Copy Detection

Unification of conformal gravity and internal interactions

Based on the observation that the dimension of the tangent space is not necessarily equal to the dimension of the corresponding curved manifold and on the known fact that gravitational theories can be formulated in a gauge theoretic way, we discuss how to describe all known interactions in a unified manner. This is achieved by enlarging the tangent group of the four-dimensional manifold to SO(2, 16), which permits the inclusion of both gauge groups, the one that describes gravity as a gauge theory as well as the SO(10) describing the internal interactions. Moreover it permits the use of both Weyl and Majorana conditions imposed on the fermions, as to avoid the duplication of fermion multiplets of SO(10) appearing in previous attempts. The gravity theory discussed in the present work is the Conformal Gravity which, after a spontaneous symmetry breaking, can lead either to Weyl Gravity or to the usual Einstein Gravity.

Unsupervised separation of nonlinearly mixed event-related potentials using manifold clustering and non-negative matrix factorization

Event-related potentials (ERPs) can quantify brain responses to reveal the neural mechanisms of sensory perception. However, ERPs often reflect nonlinear mixture responses to multiple sources of sensory stimuli, and an accurate separation of the response to each stimulus remains a challenge. This study aimed to separate the ERP into nonlinearly mixed source components specific to individual stimuli. We developed an unsupervised learning method based on clustering of manifold structures of mixture signals combined with channel optimization for signal source reconstruction using non-negative matrix factorization (NMF). Specifically, we first implemented manifold learning based on Local Tangent Space Alignment (LTSA) to extract the spatial manifold structure of multi-resolution sub-signals separated via wavelet packet transform. We then used fuzzy entropy to extract the dynamical process of the manifold structures and performed a k-means clustering to separate different sources. Lastly, we used NMF to obtain the optimal contributions of multiple channels to ensure accurate source reconstructions. We evaluated our developed approach using a simulated ERP dataset with known ground truth of two components of ERP mixture signals. Our results show that the correlation coefficient between the reconstructed source signal and the true source signal was 92.8 % and that the separation accuracy in ERP amplitude was 91.6 %. The results show that our unsupervised separation approach can accurately separate ERP signals from nonlinear mixture source components. The outcomes provide a promising way to isolate brain responses to multiple stimulus sources during multisensory perception.

STaRNet: A spatio-temporal and Riemannian network for high-performance motor imagery decoding

Brain–computer interfaces (BCIs), representing a transformative form of human–computer interaction, empower users to interact directly with external environments through brain signals. In response to the demands for high accuracy, robustness, and end-to-end capabilities within BCIs based on motor imagery (MI), this paper introduces STaRNet, a novel model that integrates multi-scale spatio-temporal convolutional neural networks (CNNs) with Riemannian geometry. Initially, STaRNet integrates a multi-scale spatio-temporal feature extraction module that captures both global and local features, facilitating the construction of Riemannian manifolds from these comprehensive spatio-temporal features. Subsequently, a matrix logarithm operation transforms the manifold-based features into the tangent space, followed by a dense layer for classification. Without preprocessing, STaRNet surpasses state-of-the-art (SOTA) models by achieving an average decoding accuracy of 83.29% and a kappa value of 0.777 on the BCI Competition IV 2a dataset, and 95.45% accuracy with a kappa value of 0.939 on the High Gamma Dataset. Additionally, a comparative analysis between STaRNet and several SOTA models, focusing on the most challenging subjects from both datasets, highlights exceptional robustness of STaRNet. Finally, the visualizations of learned frequency bands demonstrate that temporal convolutions have learned MI-related frequency bands, and the t-SNE analyses of features across multiple layers of STaRNet exhibit strong feature extraction capabilities. We believe that the accurate, robust, and end-to-end capabilities of the STaRNet will facilitate the advancement of BCIs.

Multi-source transfer learning via optimal transport feature ranking for EEG classification

Motor imagery (MI) brain-computer interface (BCI) paradigms have been extensively used in neurological rehabilitation. However, due to the required long calibration time and non-stationary nature of electroencephalogram (EEG) signals, it is challenging to obtain a substantial sample size of EEG data. Transfer learning (TL), which leverages knowledge from existing subjects to enhance the learning performance in new subjects, has been employed to solve this problem. Previous studies often extract sample features directly from the two domains for transfer learning, leading to poor features and negative transfer. To overcome this limitation, an optimal transport feature selection method was developed in this study to select the most suitable features for migration to enhance the generalization capability of the TL. This was achieved by analyzing the diagonal value similarity of the optimal transport coupling matrix. First, the Riemannian tangent space mapping method was used to map the sample covariance matrix of EEG trials from the Riemannian manifold to tangent space. Secondly, the optimal subset of EEG features was selected via the optimal transport feature selection method. Subsequently, the separate distribution alignment method was employed to minimize dissimilarities between the source and target domains. Finally, the weighted voting mechanisms were integrated for decision fusion. The proposed multi-source transfer feature learning (MSTFL) method was validated based on two public BCI Competition IV datasets. Our results demonstrated superior classification accuracy of 85.93% and 79.48%, respectively, on the two public datasets, compared to state-of-the-art models. These findings indicate the effectiveness of our proposed MSTFL model for both feature extraction and classification of MI signals.

The Wasserstein Distance for Ricci Shrinkers

Abstract Let $(M^{n},g,f)$ be a Ricci shrinker such that $\text{Ric}_{f}=\frac{1}{2}g$ and the measure induced by the weighted volume element $(4\pi )^{-\frac{n}{2}}e^{-f}dv_{g}$ is a probability measure. Given a point $p\in M$, we consider two probability measures defined in the tangent space $T_{p}M$, namely the Gaussian measure $\gamma $ and the measure $\overline{\nu }$ induced by the exponential map of $M$ to $p$. In this paper, we prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric $g_{0}$ between the measures $\overline{\nu }$ and $\gamma $, and which also elucidates the rigidity implications resulting from this estimate.

Thurston’s asymmetric metric on the space of singular flat metrics with a fixed quadrangulation

Consider a compact surface equipped with a fixed quadrangulation. One may identify each quadrangle on the surface with a Euclidean rectangle to obtain a singular flat metric on the surface with conical singularities. We call such a singular flat metric a rectangular structure. We study a metric on the space of unit area rectangular structures which is analogous to Thurston’s asymmetric metric on the Teichmüller space of a surface of finite type. We prove that the distance between two rectangular structures is equal to the logarithm of the maximum of ratios of edges of these rectangular structures. We give a sufficient condition for a path between two points of the this Teichmüller space to be geodesic and we prove that any two points of this space can be joined by a geodesic. We also prove that this metric is Finsler and give a formula for the infinitesimal weak norm on the tangent space of each point. We identify the space of unit area rectangular structures with a submanifold of a Euclidean space and we show that the subspace topology and the topology induced by the metric we introduced coincide. We show that the space of unit area rectangular structures on a surface with a fixed quadrangulation is in general not complete.

EEG functional connectivity as a Riemannian mediator: An application to malnutrition and cognition.

Mediation analysis assesses whether an exposure directly produces changes in cognitive behavior or is influenced by intermediate "mediators". Electroencephalographic (EEG) spectral measurements have been previously used as effective mediators representing diverse aspects of brain function. However, it has been necessary to collapse EEG measures onto a single scalar using standard mediation methods. In this article, we overcome this limitation and examine EEG frequency-resolved functional connectivity measures as a mediator using the full EEG cross-spectral tensor (CST). Since CST samples do not exist in Euclidean space but in the Riemannian manifold of positive-definite tensors, we transform the problem, allowing for the use of classic multivariate statistics. Toward this end, we map the data from the original manifold space to the Euclidean tangent space, eliminating redundant information to conform to a "compressed CST." The resulting object is a matrix with rows corresponding to frequencies and columns to cross spectra between channels. We have developed a novel matrix mediation approach that leverages a nuclear norm regularization to determine the matrix-valued regression parameters. Furthermore, we introduced a global test for the overall CST mediation and a test to determine specific channels and frequencies driving the mediation. We validated the method through simulations and applied it to our well-studied 50+-year Barbados Nutrition Study dataset by comparing EEGs collected in school-age children (5-11 years) who were malnourished in the first year of life with those of healthy classmate controls. We hypothesized that the CST mediates the effect of malnutrition on cognitive performance. We can now explicitly pinpoint the frequencies (delta, theta, alpha, and beta bands) and regions (frontal, central, and occipital) in which functional connectivity was altered in previously malnourished children, an improvement to prior studies. Understanding the specific networks impacted by a history of postnatal malnutrition could pave the way for developing more targeted and personalized therapeutic interventions. Our methods offer a versatile framework applicable to mediation studies encompassing matrix and Hermitian 3D tensor mediators alongside scalar exposures and outcomes, facilitating comprehensive analyses across diverse research domains.

Approaches for fast Brownian dynamics simulation with constraints

Constraints can be used to eliminate quickly fluctuating degrees of freedom in dynamical systems enabling solutions that resolve the behavior of the system with fewer time steps over long time scales. In Brownian dynamics models, these constraints lead to linear equations which represent saddle point problems describing the velocity tangent to the constraint manifold and the forces resulting from imposition of the constraint on the dynamics. Simulations of these constrained systems must solve these saddle point problems, which are often ill-conditioned and require clever preconditioning in order to facilitate large scale simulation using iterative methods like GMRES. In this work, we show that the projected conjugate gradient (PrCG) method, which applies conjugate gradients projected onto the tangent space of the constraint manifold, has certain advantages compared to preconditioned iterative methods like GMRES, without the need to devise and compute a preconditioner. Specifically, for simulations of beads interacting hydrodynamically with commonly encountered holonomic constraints, PrCG exhibits linear scaling with the number of beads similar to GMRES. However, because PrCG does not require a preconditioner it often requires less precomputational effort both in terms of computational complexity and storage, depending on the nature of the constraints. Furthermore, PrCG produces iterates which are always feasible, allowing for independent tuning of the error tolerance on the bead velocities from the error tolerance on the constraint equations. We demonstrate PrCG's efficiency by solving several example low Reynolds number systems involving rigid bodies, freely jointed chains, and immobile particles, as well as a mixed constraint problem with both a rigid body and immobile particles, comparing to GMRES and analytical solutions, when available.

Geometrically exact 3D beam theory with embedded strong discontinuities for modeling of localized failure in bending

In this work, we propose the elastoplastic model of a three-dimensional (3D) geometrically exact beam, which can capture localized bending deformation leading to a plastic hinge. This is a follow-up work to Tojaga et al. (2023), where we only studied the fracture of fiber-like beam structures that break in mode I or mode II, here extended to handle beam-like behavior with bending failure and the main difficulty of the non-vectorial character of large 3D rotations. The Reissner model is chosen for representing the large elastic and large plastic deformations of such a beam model, which leads to the multiplicative decomposition of the rotation tensor. We shown how to replace this with the corresponding additive decomposition of the rotation vector derivatives, to be performed in the material description in the tangent space of SO(3) manifold in the initial configuration. The plasticity model for which we give a more detailed implementation employs a Rankin-like multi-surface plasticity criteria, where each moment vector component (i.e. torsion or bending) is considered separately. The non-local variational formulation is proposed to deal with the softening phenomena characteristic of localized bending failure, where the fracture energy is introduced as the main parameter for softening. The discrete approximation is built in terms of the embedded-discontinuity finite element method (ED-FEM), which introduces a jump in rotation vector that can be handled at the level of a particular beam element. This kind of approach builds upon the best approximation property of the FEM discrete approximation in the energy norm and provides the best-approximation property for the dissipated energy computations, and thus optimal computational accuracy if the quantity of interest is inelastic dissipation. The computations are carried out by the operator split method, which separates the computations of global state variables (displacements and moments) from local (plastic curvature) variables. Several illustrative examples are provided to confirm an excellent performance of the proposed methodology.

Research on the Vector Space and Its Linearity

A vector space is an abelian group over a field. From the viewpoint of group theory, the tangent space is a theoretical concept. Tangent space is similar to a vector space, particularly in terms of linearity. From a field's perspective, a subfield is a subset of the original field that shares the same properties as the original field. Subfield is logically similar to a subspace of a vector space. A vector is an element of a vector space. Weight vectors can be used to describe motion, which can be applied in pattern recognition. A linear combination of weight vectors can describe a sequence of motions. Kernel graph theory treats a graph as an element of its theoretical structure. Kernel graph theory can also be applied in pattern recognition. So, basically, this article presents a perspective on applying vector space in pattern recognition. The inherent logic of vector space can be applied to both motion synthesis and graph pattern recognition. The linearity of the vector space is the main focus of this article.

Gradient Flow of the Sinai–Ruelle–Bowen Entropy

Motivated by an extension to Gallavotti–Cohen Chaotic Hypothesis, we study local and global existence of a gradient flow of the Sinai–Ruelle–Bowen entropy functional in the space of transitive Anosov maps. For the space of expanding maps on the unit circle, we equip it with a Hilbert manifold structure using a Sobolev norm in the tangent space of the manifold. Under the additional measure-preserving assumption and a slightly modified metric, we show that the gradient flow exists globally and every trajectory of the flow converges to a unique limiting map where the SRB entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow’s ordinary differential equation representation. This gradient flow has close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation.