Kernel Representation Research Articles

A fast finite-difference algorithm for solving space-fractional filtration equation with a generalised Caputo derivative

The paper considers the problem of finite-difference simulation of filtration processes on the base of a space-fractional-differential equation with a generalised Caputo derivative with respect to a function. To make calculations faster, we propose an algorithm of fractional derivative approximation which reduces the computational complexity of matrix operations that arise while performing simulations on uniform and non-uniform grids. The algorithm is based on Taylor’s series expansions of fractional integral kernel and special representation of the matrices of linear systems that arise during finite-difference approximation. Theoretical estimates of the proposed algorithm’s complexity and accuracy are presented. We describe and experimentally compare computational schemes for matrix-vector multiplication used for numerical solution of linear systems by the BicgStab algorithm and analyse factors that influence its convergence. The results of computational experiments which demonstrate an efficiency of the proposed computational scheme for solving a filtration equation with a generalised Caputo derivative with respect to a space variable on a large-sized grids are given.

Kernel Joint Sparse Representation Based on Self-Paced Learning for Hyperspectral Image Classification

By means of joint sparse representation (JSR) and kernel representation, kernel joint sparse representation (KJSR) models can effectively model the intrinsic nonlinear relations of hyperspectral data and better exploit spatial neighborhood structure to improve the classification performance of hyperspectral images. However, due to the presence of noisy or inhomogeneous pixels around the central testing pixel in the spatial domain, the performance of KJSR is greatly affected. Motivated by the idea of self-paced learning (SPL), this paper proposes a self-paced KJSR (SPKJSR) model to adaptively learn weights and sparse coefficient vectors for different neighboring pixels in the kernel-based feature space. SPL strateges can learn a weight to indicate the difficulty of feature pixels within a spatial neighborhood. By assigning small weights for unimportant or complex pixels, the negative effect of inhomogeneous or noisy neighboring pixels can be suppressed. Hence, SPKJSR is usually much more robust. Experimental results on Indian Pines and Salinas hyperspectral data sets demonstrate that SPKJSR is much more effective than traditional JSR and KJSR models.

Mixture of Gaussian Blur Kernel Representation for Blind Image Restoration

The use of blind image restoration, sharpness of edges may frequently be restored using previous information from a picture. De-blurring is the technique of taking out blurring flaws of the steady photographs, including motion or defocus aberration-related blur. the appearance of fast-moving the appearance of fast-moving entities flashing in still images flashing in a still photograph is known as motion blur. When an image is blurred using a Gaussian function, the result is a Gaussian blur. The employment of different sparse priors, either for the implicit photos or the motion blur kernels, contributes to the success of contemporary single-image approaches. De-blurring is the technique of taking out blurring flaws of the steady photographs, including motion or defocus aberration-related blur. The apparent flashing of quickly moving item in a static photograph is known as motion blur. When a picture is blurred utilizing a Gaussian function, the result is a Gaussian blur. The employment of different sparse priors, either for the latent photos or the motion blur kernels, contributes to the success of contemporary single-image approaches. On digital datasets, KSR also discovers effective kernel matrix approximation to hasten blurring and provide effective de-blur performances. The licence plate, which serves as the vehicle's distinctive identifier, is an important indicator of speeding or hit-and-run cars. However, the image of a fast-moving car taken by a security camera is usually blurred and not even humanly discernible.These observed plate pictures are frequently poor resolution and have significant edge information lost, which presents a significant challenge to the current blind deblurring techniques. The blur kernel may be thought of as a linear uniform convolution and parametrically modelled with angle and length for licence plate picture blurring brought on by rapid movement. This research suggests unique technique for locating the blur kernel based on sparse representation. We determine the angle of the kernel by looking at the sparse representation coefficients of the restored picture because the retrieved photo has the highest sparse representation when the kernel angle coincides with the real motion angle. Afterwards, using Radon transform in Fourier domain, we estimate the size of the motion kernel. Even when the licence plate is impossible for a person to read, our system handles big motion blur rather effectively. We assess our method using actual photographs and contrast it with a number of well-known cutting-edge blind image deblurring techniques. Experimental findings show that our suggested technique is superior in terms of efficiency and resilience.

An inherently consistent reproducing kernel gradient smoothing framework toward efficient Galerkin meshfree formulation with explicit quadrature

A reproducing kernel gradient smoothing framework with explicit quadrature rules is proposed for efficient Galerkin meshfree formulation. In this framework, the meshfree smoothed gradients are formulated via a reproducing kernel representation of the standard gradients of field variables. It is interesting to find that this reproducing kernel construction of smoothed gradients of meshfree shape functions enables an identical satisfaction of the integration constraint derived from Galerkin meshfree formulation. In other words, the integration consistency is an inherent property of the proposed reproducing kernel gradient smoothing methodology regardless of integration schemes. Consequently, without violation of the integration constraint, the conventional normal low order Gauss quadrature rules in finite element analysis now can be used to properly integrate the meshfree stiffness matrix simply through replacing the standard meshfree gradients with the reproducing kernel smoothed gradients at Gauss points. In order to efficiently compute the reproducing kernel smoothed gradients, a set of explicit quadrature rules referring to 2D triangular and 3D tetrahedral integration cells are systematically developed. The total number of sample points associated with these quadrature rules is minimized from a global point of view through introducing an equivalent number of sample points, which implies as many as sample points are shared by neighboring integration cells. The proposed methodology recovers the stabilized conforming nodal integration when linear basis function is employed, while the focus of the present work is multidimensional higher order basis functions such as quadratic and cubic ones. Superior convergence, accuracy as well as efficiency performances of the proposed reproducing kernel gradient smoothing framework are thoroughly demonstrated by a series of numerical examples.

Three- and four-body kaonic nuclear states: Binding energies and widths

Using separable potentials for $\bar{K}N-\pi\Sigma$ interaction, we investigated four-body kaonic nuclear systems such as $K^{-}ppn$ and $K^{-}K^{-}pp$, with the Faddeev AGS method in the momentum representation. The Faddeev calculations are based on the quasi-particle method and the method of the energy dependent pole expansion was used to obtain the separable representation for the integral kernels in the three- and four-body equations. Different types of $\bar{K}N-\pi\Sigma$ potentials based on phenomenological and chiral SU(3) approach are used and it was shown that the kaonic nuclear systems under consideration are tightly bound.

A representation of the transmutation kernels for the Schrödinger operator in terms of eigenfunctions and applications

The representations of the kernels of the transmutation operator and of its inverse relating the one-dimensional Schrödinger operator with the second derivative are obtained in terms of the eigenfunctions of a corresponding Sturm–Liouville problem. Since both series converge slowly and in general only in a certain distributional sense we find a way to improve these expansions and make them convergent uniformly and absolutely by adding and subtracting corresponding terms. A numerical illustration of the obtained results is given.

Quality-Driven Kernel Projection to Latent Structure Model for Nonlinear Process Monitoring

A novel quality-driven kernel projection to latent structure (QKPLS) modeling scheme is proposed for concurrent quality-related and process-fault detection for nonlinear processes. Process data are initially mapped into a high-dimensional feature space by nonlinear mapping. The mapped data in the feature space are then projected by kernel representation into a process-dominant subspace that captures the main process variance and a process-residual subspace orthogonal to the process-dominant subspace. On the basis of the relationship with quality variables, the process-dominant subspace is further decomposed into two orthogonal subspaces, namely, a quality-related subspace that maximizes the covariance between the subspace and the quality variables and a quality-residual subspace orthogonal to the quality-related subspace. Afterward, three orthogonal subspaces are obtained, and monitoring statistics are established to achieve concurrent quality-related and process-fault detection. The application examples on a numerical example and Tennessee Eastman process verify the effectiveness of the QKPLS-based monitoring scheme.

Internal Emotion Classification Using EEG Signal With Sparse Discriminative Ensemble

Among various physiological signal acquisition methods for the study of the human brain, EEG (Electroencephalography) is more effective. EEG provides a convenient, non-intrusive, and accurate way of capturing brain signals in multiple channels at fine temporal resolution. We propose an ensemble learning algorithm for automatically computing the most discriminative subset of EEG channels for internal emotion recognition. Our method describes an EEG channel using kernel-based representations computed from the training EEG recordings. For ensemble learning, we formulate a graph embedding linear discriminant objective function using the kernel representations. The objective function is efficiently solved via sparse non-negative principal component analysis and the final classifier is learned using the sparse projection coefficients. Our algorithm is useful in reducing the amount of data while improving computational efficiency and classification accuracy at the same time. The experiments on publicly available EEG dataset demonstrate the superiority of the proposed algorithm over the compared methods.

Human Action Recognition Using Multilevel Depth Motion Maps

The advent of depth sensors opens up new opportunities for human action recognition by providing depth information. The main purpose of this paper is to present an effective method for human action recognition from depth images. A multilevel frame select sampling (MFSS) method are proposed to generate three levels of temporal samples from the input depth sequences first. Then, the proposed motion and static mapping (MSM) method is used to obtain the representation of MFSS sequences. After that, this paper exploits the block-based LBP feature extraction approach to extract features information from the MSM. Finally, the fisher kernel representation is applied to aggregate the block features, which is then combined with the kernel-based extreme learning machine classifier. The developed framework is evaluated on three public datasets captured by depth cameras. The experimental results demonstrate the great performance compared with the existing approaches.

Schizophrenia Classification Using fMRI Data Based on a Multiple Feature Image Capsule Network Ensemble

Automatic diagnosis and classification of schizophrenia based on functional magnetic resonance imaging (fMRI) data have attracted increasing attention in recent years. Most previous studies abstracted highly compressed functional features from the view of brain science and fed them into shallow classifiers for this purpose. However, their classification performance in practical applications is unstable and unsatisfactory. As an acute psychotic disorder, schizophrenia shows functional complexity in fMRI data. Therefore, additional features and deep classification methods are needed to improve classification performance. In this study, we propose a multiple feature image capsule network ensemble approach for schizophrenia classification. The proposed approach proceeds in three steps: 1) extracting multiple image features from the perspective of linear sparse representation, nonlinear multiple kernel representation, and function connection of brain areas respectively; 2) feeding these image features into three specially designed independent capsule networks for classification; 3) obtaining the final results by fusing the outputs of these three deep capsule network using a ensemble approach. To further improve the classification performance, we design a optimization model of maximizing the square of correlation coefficients and propose a weighted ensemble technology based on this model, which is mathematically proved to be solved as a eigenvalue decomposition problem in certain case. Finally, the proposed approach is implemented and evaluated on the schizophrenia fMRI dataset from COBRE, UCLA and WUSTL. From the experimental results, we conclude that the proposed method outperforms some current methods and further improves the accuracy of schizophrenia classification.

Lossless Systems Storage Function: New Results and Numerically Stable and Non-Iterative Computational Methods

In this paper, we formulate and prove new results in the context of storage functions for lossless systems: we use these results to propose new algorithms to compute the storage function. The computation of the storage function for the lossless case is not possible using conventional algebraic Riccati equation-based algorithms, though the storage function itself is well-defined. This is because a certain “regularity condition” on the feedthrough term in the i/s/o representation of the lossless system does not hold. We formulate new results about the storage function matrix for the lossless case and use them to propose non-iterative and stable algorithms to compute the storage function directly from different representations of the given system, namely, a kernel representation, transfer function, and the i/s/o representation of the system. Across the methods, for randomly generated transfer functions, we compare: 1) the computational effort (in flops); 2) the computation time using numerical experiments; and 3) the computational error.

Spatially homogeneous copulas

We consider spatially homogeneous copulas, i.e. copulas whose corresponding measure is invariant under a special transformations of $$[0,1]^2$$, and we study their main properties with a view to possible use in stochastic models. Specifically, we express any spatially homogeneous copula in terms of a probability measure on [0, 1) via the Markov kernel representation. Moreover, we prove some symmetry properties and demonstrate how spatially homogeneous copulas can be used in order to construct copulas with surprisingly singular properties. Finally, a generalization of spatially homogeneous copulas to the so-called (m, n)-spatially homogeneous copulas is studied and a characterization of this new family of copulas in terms of the Markov $$*$$-product is established.

On a Series Representation for Integral Kernels of Transmutation Operators for Perturbed Bessel Equations

A representation for the kernel of the transmutation operator relating a perturbed Bessel equation to the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure. New properties of the transmutation kernel are established. A new representation of a regular solution of a perturbed Bessel equation is given, which admits a uniform error bound with respect to the spectral parameter for partial sums of the series. A numerical illustration of the application of the obtained result to solve Dirichlet spectral problems is presented.

On optimal stopping of multidimensional diffusions

This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the d-dimensional Wiener process. We first obtain some verification theorems for diffusions, based on the Green kernel representation of the value function. Specializing to the multidimensional Wiener process, we apply the Martin boundary theory to obtain a set of tractable integral equations involving only harmonic functions that characterize the stopping region of the problem in the bounded case. The approach is illustrated through the optimal stopping problem of a d-dimensional Wiener process with a positive definite quadratic form reward function.

A Fault Detection Approach for Nonlinear Systems Based on Data-Driven Realizations of Fuzzy Kernel Representations

This paper is devoted to the data-driven fault detection of nonlinear systems. For our purpose, the definition of Takagi–Sugeno fuzzy data-driven forms of kernel representations for nonlinear systems is introduced first, which builds the basis of our work. The major contributions consist of two parts. In the first part, a data-driven method for fuzzy process modeling is proposed, and associated with it, some modeling issues are addressed with the aid of the so-called randomized algorithm technique in the probabilistic framework. It is followed by a data-driven realization of fuzzy kernel representation and its implementation in the fault detection system design. To link the data-driven methods to the well-established observer-based fault detection approaches, the recursive form of the fuzzy kernel representation is proposed. In the second part, the fuzzy-observer-based fault detection design scheme is investigated based on the recursive fuzzy kernel representation. The main results of our study are illustrated by an experimental study on the laboratory setup of a three-tank system.

Optimal existence classes and nonlinear-like dynamics in the linear heat equation in [formula omitted

We analyse the behaviour of solutions of the linear heat equation in Rd for initial data in the classes Mε(Rd) of Radon measures with ∫Rde−ε|x|2d|u0|<∞. We show that these classes are optimal for local and global existence of non-negative solutions: in particular M0(Rd):=∩ε>0Mε(Rd) consists of those initial data for which a solution of the heat equation can be given for all time using the heat kernel representation formula. We prove existence, uniqueness, and regularity results for such initial data, which can grow rapidly at infinity, and then show that they give rise to properties associated more often with nonlinear models. We demonstrate the finite-time blowup of solutions, showing that the set of blowup points is the complement of a convex set, and that given any closed convex set there is an initial condition whose solutions remain bounded precisely on this set at the ‘blowup time’. We also show that wild oscillations are possible from non-negative initial data as t→∞ and that one can prescribe the behaviour of u(0,t) to be any real-analytic function γ(t) on [0,∞).

Fourier Feature Approximations for Periodic Kernels in Time-Series Modelling

Gaussian Processes (GPs) provide an extremely powerful mechanism to model a variety of problems but incur an O(N3) complexity in the number of data samples. Common approximation methods rely on what are often termed inducing points but still typically incur an O(NM2) complexity in the data and corresponding inducing points. Using Random Fourier Feature (RFF) maps, we overcome this by transforming the problem into a Bayesian Linear Regression formulation upon which we apply a Bayesian Variational treatment that also allows learning the corresponding kernel hyperparameters, likelihood and noise parameters. In this paper we introduce an alternative method using Fourier series to obtain spectral representations of common kernels, in particular for periodic warpings, which surprisingly have a convergent, non-random form using special functions, requiring fewer spectral features to approximate their corresponding kernel to high accuracy. Using this, we can fuse the Random Fourier Feature spectral representations of common kernels with their periodic counterparts to show how they can more effectively and expressively learn patterns in time-series for both interpolation and extrapolation. This method combines robustness, scalability and equally importantly, interpretability through a symbolic declarative grammar that is both functionally and humanly intuitive — a property that is crucial for explainable decision making. Using probabilistic programming and Variational Inference we are able to efficiently optimise over these rich functional representations. We show significantly improved Gram matrix approximation errors, and also demonstrate the method in several time-series problems comparing other commonly used approaches such as recurrent neural networks.

Commutative POV-Measures: from the Choquet Representation to the Markov Kernel and Back

We study two relevant characterizations of a commutative positive operator valued measure (POVM) F. The first one is a Choquet type of an integral representation. It introduces a measure ν on the space of the projection valued measures (PVMs) and describes F as an integral over this space. The second one represents a commutative POVM F as the randomization of a single PVM E by means of a Markov kernel μ. We show that one can be derived from the other. We also elaborate upon some previous results on Choquet’s representation of Markov kernels and find a functional relationship between ν and μ. Finally, we analyze some relevant particular cases and provide some physically relevant examples which include the unsharp position observables.

Efron’s monotonicity property for measures on [formula omitted

In this paper, we prove some kernel representations for the covariance of two functions taken on the same random variable and we deduce kernel representations for some functionals of a continuous one-dimensional measure. Then we apply these formulas to extend Efron’s monotonicity property, given in Efron (1965) and valid for independent log-concave measures, to the case of general measures on R2. The new formulas are also used to derive some further quantitative estimates in Efron’s monotonicity property.

A Data-Driven Computation Method for the Gap Metric and the Optimal Stability Margin

Gap metric and stability margin have been proven as important model-based analysis tools for linear time-invariant (LTI) systems. Due to the need for an accurate model, both the gap metric and the optimal stability margin are mostly used for offline analysis. In online analysis, accurate models are rarely available, whereas measurement data could be easily obtained from the systems under consideration. Therefore, in this paper, an approach toward the calculation of the gap metric and the optimal stability margin based on the available measurement data of LTI systems is proposed. A data-driven realization of the so-called stable kernel and stable image representation serves as the foundation of this framework.