Symmetric Positive Definite Matrices Research Articles

Explicit Runge–Kutta Numerical Manifold Method for Solving the Burgers’ Equation via the Hopf–Cole Transformation

This paper presents an efficient numerical manifold method for solving the Burgers’ equation. To improve accuracy and streamline the solution process, we apply a nonlinear function transformation technique that reformulates the original problem into a linear diffusion equation. We utilize a dual cover mesh along with an explicit multi-step time integration method for spatial and temporal discretization, respectively. Constant cover functions are employed across mathematical covers, interconnected by a linear weight function for each manifold element. The full discretization formulation is derived using the Galerkin weak form. To efficiently compute the inverse of the symmetric positive definite mass matrix, we employ the Crout algorithm. The performance and convergence of our method are rigorously evaluated through several benchmark numerical tests. Extensive comparisons with exact solutions and alternative methods demonstrate that our approach delivers an accurate, stable, and efficient computational scheme for the Burgers’ equation.

Incremental Classification for High-Dimensional EEG Manifold Representation Using Bidirectional Dimensionality Reduction and Prototype Learning.

In brain-computer interface (BCI) systems, symmetric positive definite (SPD) manifold within Riemannian space has been frequently utilized to extract spatial features from electroencephalogram (EEG) signals. However, the intrinsic high dimensionality of SPD matrices introduces too much computational burden to hinder the real-time applications of such BCI, especially in handling dynamic tasks, like incremental learning. Directly reducing the dimensionality of SPD matrices with conventional dimensionality reduction (DR) methods will alter the fundamental properties of SPD matrices. Moreover, current DR methods for incremental learning always necessitate retaining old data to update their representations under new mapping. To this end, a bidirectional two-dimensional principal component analysis for SPD manifold (B2DPCA-SPD) is proposed to reduce the dimensionality of SPD matrices, in such way that the reduced matrices remain on SPD manifold. Afterwards, the B2DPCA-SPD is extended to adapt to incremental learning tasks without saving old data. The incremental B2DPCA-SPD can be seamlessly integrated with the matrix-formed growing neural gas network (MF-GNG) to achieve an incremental EEG classification, where the new low-dimensional representations of the prototypes in old classifiers can be easily recalculated with the updated projection matrix. Extensive experiments are conducted on two public datasets to perform the EEG classification. The results demonstrate that our method significantly reduces computation time by 38.53% and 35.96%, and outperforms conventional methods in classification accuracy by 4.21% to 19.59%.

New error estimates for the conjugate gradient method

The conjugate gradient method is the default iterative method for the solution of linear systems of equations with a large symmetric positive definite matrix A. The development of techniques for estimating the norm of the error in iterates computed by this method has received considerable attention. Available methods for bracketing the A-norm of the error evaluate pairs of Gauss and Gauss-Radau quadrature rules to determine lower and upper bounds. The latter methods require a user to allocate a node (the Radau node) between the origin and the smallest eigenvalue of the system matrix. The determination of such a node generally demands further computations to estimate the location of the smallest eigenvalue. An approach that avoids the need for a lower bound of the smallest eigenvalue is to replace the Gauss-Radau quadrature rule by an anti-Gauss rule; see, e.g., Golub and Meurant (1997), Golub and Meurant (2010), Golub and Strakoš (1994), Meurant (1997), Meurant (1999). However, this approach may sometimes yield inaccurate error norm estimates. This paper proposes the use of pairs of Gauss and associated optimal averaged Gauss quadrature rules to estimate the A-norm of the error in iterates determined by the conjugate gradient method.

Log-Cholesky filtering of diffusion tensor fields: Impact on noise reduction

Diffusion tensor imaging (DTI) is a powerful neuroimaging technique that provides valuable insights into the microstructure and connectivity of the brain. By measuring the diffusion of water molecules along neuronal fibers, DTI allows the visualization and study of intricate networks of neural pathways.DTI is a noise-sensitive method, where a low signal-to-noise ratio (SNR) results in significant errors in the estimated tensor field. Tensor field regularization is an effective solution for noise reduction.Diffusion tensors are represented by symmetric positive-definite (SPD) matrices. The space of SPD matrices may be viewed as a Riemannian manifold after defining a suitable metric on its tangent bundle. The Log-Cholesky metric is a recently developed concept with advantages over previously defined Riemannian metrics, such as the affine-invariant and Log-Euclidean metrics. The utility of the Log-Cholesky metric for tensor field regularization and noise reduction has not been investigated in detail.This manuscript provides a quantitative investigation of the impact of Log-Cholesky filtering on noise reduction in DTI. It also provides sufficient details of the linear algebra and abstract differential geometry concepts necessary to implement this technique as a simple and effective solution to filtering diffusion tensor fields.

Metric meta-learning and intrinsic Riemannian embedding for writer independent offline signature verification

Offline signature verification necessitates the involvement of machine learning visual recognition techniques. Efficient signature e-verifiers in machine learning and data analysis methods often inherently assume that the visual representation of the signature data can be modeled by vectors of Euclidean nature. However, as scientists become increasingly aware, ignoring an intrinsically non-Euclidean, or geometric, structure of the data can lead to sub-optimal results. The objective of this research is to enhance offline signature verification by incorporating non-Euclidean representations, particularly symmetric positive definite matrices (SPD). Specifically, this work proposes a geometrically inspired solution to the challenging task of verifying a questioned signature sample given a reference set of template signatures. Thus, a pair of covariance matrices is intrinsically embedded into a vector, denoted as the Embedded Riemannian Pyramid (ERP) by exploring a set of pairwise SPD measures, between sub-SPD subsets originating from a pair of covariance matrices. In conclusion, the SigmML, a meta-optimization network in the space of the SPD matrices, is introduced as a novel metric learning framework with low-dimensional ERP embeddings as inputs. Evaluation is conducted under two blind scenarios, namely blind intra-dataset and cross-lingual testing, utilizing a multitude of signature datasets or different origins.

A symmetric interior-penalty discontinuous Galerkin isogeometric analysis spatial discretization of the self-adjoint angular flux form of the neutron transport equation

This paper presents the first application of a symmetric interior-penalty discontinuous Galerkin isogeometric analysis (SIP-DG-IGA) spatial discretization to the self-adjoint angular flux (SAAF) form of the multi-group neutron transport equation. The penalty parameters are determined, for general element types, from a mathematically rigorous coercivity analysis of the bilinear form. The proposed scheme produces a compact spatial discretization stencil. It also yields symmetric positive-definite (SPD) matrices, which can be efficiently solved using pre-conditioned conjugate gradient (PCG) solution algorithms. The proposed discretization scheme is verified using the method of manufactured solutions (MMS) and several nuclear reactor physics benchmark verification test cases. For sufficiently smooth elliptic problems, the proposed spatial discretization can exploit higher-order continuity, or k-refinement, of the NURBS basis to consistently yield greater numerical accuracy per degree of freedom (DoF) than standard h-refinement. Since this is a discontinuous scheme, it can also accurately model significant changes in the neutron scalar flux that may occur near the material interfaces of heterogeneous problems.

Classification of BCI-EEG Based on the Augmented Covariance Matrix.

Electroencephalography signals are recorded as multidimensional datasets. We propose a new framework based on the augmented covariance that stems from an autoregressive model to improve motor imagery classification. From the autoregressive model can be derived the Yule-Walker equations, which show the emergence of a symmetric positive definite matrix: the augmented covariance matrix. The state-of the art for classifying covariance matrices is based on Riemannian Geometry. A fairly natural idea is therefore to apply this Riemannian Geometry based approach to these augmented covariance matrices. The methodology for creating the augmented covariance matrix shows a natural connection with the delay embedding theorem proposed by Takens for dynamical systems. Such an embedding method is based on the knowledge of two parameters: the delay and the embedding dimension, respectively related to the lag and the order of the autoregressive model. This approach provides new methods to compute the hyper-parameters in addition to standard grid search. The augmented covariance matrix performed ACMs better than any state-of-the-art methods. We will test our approach on several datasets and several subjects using the MOABB framework, using both within-session and cross-session evaluation. The improvement in results is due to the fact that the augmented covariance matrix incorporates not only spatial but also temporal information. As such, it contains information on the nonlinear components of the signal through the embedding procedure, which allows the leveraging of dynamical systems algorithms. These results extend the concepts and the results of the Riemannian distance based classification algorithm.

Neural network-based distributed consensus tracking control for uncertain Euler–Lagrange systems over directed topologies

This paper proposes a neural network-based robust control approach for driving disturbed Euler–Lagrange systems (e.g., robot manipulator) in a directed network to perform consensus tracking of a heterogeneous leader’s output signal. The parameter matrices and external disturbances in each follower’s Euler–Lagrange dynamics are unmeasurable, thus we unify them into one lumped uncertainty term. Neural networks are then employed to online estimate these uncertainties, with adaptive update laws ensuring the boundedness of the estimation errors. Subsequently, a set of distributed observers are developed to adaptively estimate the leader’s states as well as the unidentified parameter vector and output matrix in the leader’s model. With their help, the robust cooperative controller is designed. Its efficacy on directed networks is theoretically justified by designing a Lyapunov function with a special structure. This function produces symmetric positive definite matrices when its first derivative terms interact with the specified parameters in the observer and Laplacian matrices for directed graphs. Finally, numerical simulations based on two-link robot manipulators are carried out to verify that the tracking errors converge to zero when applying the neural network-based robust controller.

Personal identification system based on multidimensional electroencephalographic signals

Personal authentication using electroencephalographic (EEG) signals, is one of the important applications in brain computer interface (BCI). In this work we investigate the use of EEG signals as a biometric trait. Multidimensional EEG signals were represented as symmetric positive-definite (SPD) matrices on a Riemannian manifold. Two experiments are performed in the first; we use minimum distance to Riemannian mean (MDRM) as a classifier. In the second; SPD matrices are vectorized, and the generated vectors are used to train various machine learning (ML) classifiers. MDRM classifier achieved a correct recognition rate (CRR) of 96.92% , while ML classifiers achieved CRR from 95.39% to 99.45%.

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

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

From constraints fusion to manifold optimization: A new directional transport manifold metaheuristic algorithm

The ascent of geometry-based models and methodologies, exemplified by geometric deep learning and manifold numerical optimization algorithms, has inaugurated a novel domain across various applications that grapple with geometric data complexities, such as electroencephalogram signals represented by symmetric positive definite matrix manifold, hierarchical data represented by hyperbolic manifold. The imperative fuels this inevitable paradigm shift to encapsulate the intricacies and richness inherent in data, areas where traditional methods prove inadequate. While metaheuristic algorithms are renowned for their versatile adaptability across applications, offering practical solutions within reasonable timeframes. However, the conventional metaheuristic algorithms fail on manifold applications with meaningless solutions. From an extrinsic optimization perspective, we treat manifold optimization problems as general optimization problems with multiple fused constraints that limit the optimization path to the manifold. This study pioneered the proposal and implementation of a metaheuristic manifold optimization, introducing a novel directional transport operator to rectify previously identified issues. Through experimentation across five sets of 25 problems, comparing against five algorithms, including both gradient-free and gradient-dominant counterparts, our proposed algorithm emerges as the optimal performer within the gradient-free category, demonstrating competitiveness even against gradient-dominant algorithms. Furthermore, we applied the proposed algorithm to the robot dynamic manipulation problem, achieving a close-optimal solution that eludes gradient-dominant approaches. This paper delves into the inherent capabilities and establishes the generalization of a metaheuristic algorithm within non-Euclidean functional landscapes. The source code will be available at https://github.com/lingping-fuzzy/metaheuristic-manifold-optimization.

Natural gas pipeline leak diagnosis based on manifold learning

Natural gas pipeline leakage is a common safety hazard, which can have a great impact on the economy and the environment. This paper proposed a novel manifold learning-enabled feature extraction method for natural gas pipeline leakage diagnosis. Firstly, the natural gas pipeline working condition signal is decomposed and denoised by Variational mode decomposition (VMD). Secondly, the denoised pipeline signals were constructed into a form expressed by the Symmetric positive definite matrix (SPD) using the VMD reconstruction technique, and the geodesic distance measurement method was applied to the SPD matrix to make the data located on the SPD manifold. Then feature extraction is carried out by Local linear embedding (LLE) method based on asymmetric distance. Finally, pattern recognition of the features extracted in this paper by Support vector machine (SVM) can achieve 100% recognition accuracy. By enabling faster and more accurate leak detection, the method minimizes gas loss, as well as mitigating the environmental risks caused by this potent greenhouse gas.

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.

A Novel Weak-Form Plane Frame Quadrature Element and its Application to Bridge Analysis

A novel weak-form plane frame quadrature element (WPFQE) is developed. A complex structure can be divided into several non-homogeneous (or homogeneous) elements with large size. The Chebyshev–Lobatto differential quadrature dealing with differentiation and the Gauss–Lobatto quadrature dealing with integration have the same integration points including the ends of each element. Based on the energy variational principle, the diagonal mass matrix and positive definite real symmetric stiffness matrix for the element can be obtained. The unknown quantities at the internal quadrature points of the element can be represented by those at the endpoints of the element based on the principle of static equal effect anterior to the development of global matrices, which significantly reduces the sizes of modified element matrices while maintaining sufficient accuracy, thereby obtaining the small-sized global matrices. Assembly of elements can be performed efficiently just as that in the finite element (FE) method. In the case study, the static and dynamic performances of a three-span high-pier rigid-frame bridge with variable cross-section are studied by applying the proposed method. The results indicate that the orders of structural matrices and calculation time obtained by the WPFQE method are much smaller than those of the FE method.

Kernel Angle Dependence Measures in Metric Spaces

Measuring and testing dependence between data in separable metric spaces is of great importance in modern statistics. Most existing work relied on the distance between random variables, which inevitably required the moment conditions to guarantee the distance is well-defined. Based on the geometry element “angle”, we develop a novel class of nonlinear dependence measures for data in metric space that can avoid such conditions. Specifically, by making use of the reproducing kernel Hilbert space equipped with Gaussian measure, we introduce kernel angle covariances that can be applied to various types of data, including low dimensional vector, high dimensional vectors, non-Euclidean data like symmetric positive definite matrices, and compositional data. We estimate kernel angle covariances based on U-statistic and establish the corresponding independence tests via gamma approximation. Our kernel angle independence tests, imposing no-moment conditions on kernels, are robust with heavy-tailed random variables. We conduct comprehensive simulation studies and apply our proposed methods to a facial recognition task. Our kernel angle covariances-based tests show remarkable performances in dealing with image data. All the codes and proofs are included in the supplementary materials.

Robust speckle covariance matrix estimation of sea clutter based on spectral symmetry

Speckle covariance matrix estimation plays an important role in adaptive target detection of maritime radars. Spatial heterogeneity of sea clutter incurs insufficient secondary data in estimation, which degrades detection performance. It is an effective way of estimation improvement to exploit the Doppler spectrum knowledge of sea clutter for structural dimension reduction of the speckle covariance matrix. This paper proposes a speckle covariance matrix estimation method based on the spectral symmetry knowledge of sea clutter, indicating that the spectrum of sea clutter is symmetric with respect to the Doppler offset. First of all, the spectral symmetry of sea clutter is analyzed by using the measured data from several opened databases and the results show that most of the data excluding sea spikes have symmetric Doppler spectra. Secondly, the spectral symmetry constrains the speckle covariance matrix into the Hadamard product of a special rank-one Toeplitz matrix and a real positive definite symmetric matrix. The degrees of freedom are reduced to almost half of that of a Hermitian matrix and thus fewer secondary data are required. Thirdly, a robust iterative maximum likelihood estimator is proposed to estimate the speckle covariance matrix with structural constraint and is free of the clutter texture distribution. The performance of the estimator is verified by the measured data and the results show that it is superior to traditional estimators, particularly in fewer secondary data.

SPD Manifold Deep Metric Learning for Image Set Classification.

By characterizing each image set as a nonsingular covariance matrix on the symmetric positive definite (SPD) manifold, the approaches of visual content classification with image sets have made impressive progress. However, the key challenge of unhelpfully large intraclass variability and interclass similarity of representations remains open to date. Although, several recent studies have mitigated the two problems by jointly learning the embedding mapping and the similarity metric on the original SPD manifold, their inherent shallow and linear feature transformation mechanism are not powerful enough to capture useful geometric features, especially in complex scenarios. To this end, this article explores a novel approach, termed SPD manifold deep metric learning (SMDML), for image set classification. Specifically, SMDML first selects a prevailing SPD manifold neural network (SPDNet) as the backbone (encoder) to derive an SPD matrix nonlinear representation. To counteract the degradation of structural information during multistage feature embedding, we construct a Riemannian decoder at the end of the encoder, trained by a reconstruction error term (RT), to induce the generated low-dimensional feature manifold of the hidden layer to capture the pivotal information about the visual data describing the imaged scene. We demonstrate through theory and experiments that it is feasible to replace the Riemannian metric with Euclidean distance in RT. Then, the ReCov layer is introduced into the established Riemannian network to regularize the local statistical information within each input feature matrix, which enhances the effectiveness of the learning process. The theoretical analysis of the activation function used in the ReCov layer in terms of continuity and conditions for generating positive definite matrices is beneficial for network design. Inspired by the fact that the single cross-entropy loss used for training is unable to effectively parse the geometric distribution of the deep representations, we finally endow the suggested model with a novel metric learning regularization term. By explicitly incorporating the encoding and processing of the data variations into the network learning process, this term can not only derive a powerful Riemannian representation but also train an effective classifier. The experimental results show the superiority of the proposed approach on three typical visual classification tasks.

On the impact of spatial covariance matrix ordering on tile low‐rank estimation of Matérn parameters

AbstractSpatial statistical modeling involves processing an symmetric positive definite covariance matrix, where denotes the number of locations. However, when is large, processing this covariance matrix using traditional methods becomes prohibitive. Thus, coupling parallel processing with approximation can be an elegant solution by relying on parallel solvers that deal with the matrix as a set of small tiles instead of the full structure. The approximation can also be performed at the tile level for better compression and faster execution. The tile low‐rank (TLR) approximation has recently been used to compress the covariance matrix, which mainly relies on ordering the matrix elements, which can impact the compression quality and the efficiency of the underlying solvers. This work investigates the accuracy and performance of location‐based ordering algorithms. We highlight the pros and cons of each ordering algorithm and give practitioners hints on carefully choosing the ordering algorithm for TLR approximation. We assess the quality of the compression and the accuracy of the statistical parameter estimates of the Matérn covariance function using TLR approximation under various ordering algorithms and settings of correlations through simulations on irregular grids. Our conclusions are supported by an application to daily soil moisture data in the Mississippi Basin area.

Differentiable microstructures design via anisotropic thermal diffusion

We propose a novel method to design differentiable microstructures. Central to our algorithm is a new representation of the mapping from the parameters to microstructures, formulated as the anisotropic thermal diffusion. A metric field governs the anisotropic diffusion. The metric associated with each point is represented as a 2 × 2 symmetric positive definite matrix that becomes the design variable. To alleviate the difficulties caused by symmetric positive definite constraints, we perform the singular value decomposition of the metric matrix so that the design variable includes a rotation angle and a diagonal matrix. Then, the positive definiteness is converted to requiring the two diagonal entries of the diagonal matrix to be positive, which is easier to deal with. The effectiveness of our algorithm is demonstrated through evaluations and comparisons over various examples.

Topology of surface electromyogram signals: hand gesture decoding on Riemannian manifolds

Objective. Decoding gestures from the upper limb using noninvasive surface electromyogram (sEMG) signals is of keen interest for the rehabilitation of amputees, artificial supernumerary limb augmentation, gestural control of computers, and virtual/augmented realities. We show that sEMG signals recorded across an array of sensor electrodes in multiple spatial locations around the forearm evince a rich geometric pattern of global motor unit (MU) activity that can be leveraged to distinguish different hand gestures. Approach. We demonstrate a simple technique to analyze spatial patterns of muscle MU activity within a temporal window and show that distinct gestures can be classified in both supervised and unsupervised manners. Specifically, we construct symmetric positive definite covariance matrices to represent the spatial distribution of MU activity in a time window of interest, calculated as pairwise covariance of electrical signals measured across different electrodes. Main results. This allows us to understand and manipulate multivariate sEMG timeseries on a more natural subspace—the Riemannian manifold. Furthermore, it directly addresses signal variability across individuals and sessions, which remains a major challenge in the field. sEMG signals measured at a single electrode lack contextual information such as how various anatomical and physiological factors influence the signals and how their combined effect alters the evident interaction among neighboring muscles. Significance. As we show here, analyzing spatial patterns using covariance matrices on Riemannian manifolds allows us to robustly model complex interactions across spatially distributed MUs and provides a flexible and transparent framework to quantify differences in sEMG signals across individuals. The proposed method is novel in the study of sEMG signals and its performance exceeds the current benchmarks while being computationally efficient.