Mercer's Theorem Research Articles

Geospatial data for peer-to-peer communication among autonomous vehicles using optimized machine learning algorithm

The transportation infrastructure of the future will be based on autonomous vehicles. When it comes to transportation, both emerging and established nations are keen on perfecting systems based on autonomous vehicles. Transportation authorities in the United States report that driver error accounts for over 60% of all accidents each year. Almost everywhere in the world is the same. Since the idea of self-driving cars involves a fusion of hardware and software. Despite the rapid expansion of the software business and the widespread adoption of cutting-edge technologies like AI, ML, Data Science, Big Data, etc. However, the identification of natural disasters and the exchange of data between vehicles present the greatest hurdle to the development of autonomous vehicles. The suggested study primarily focused on data cleansing from the cars, allowing for seamless interaction amongst autonomous vehicles. This study's overarching goal is to look at creating a novel kind of Support Vector Machine kernel specifically for P2P networks. To meet the kernel constraints of Mercer's theorem, a newly proposed W-SVM (Weighted-SVM) kernel was produced by using an appropriately converted weight vector derived through hybrid optimization. Given the advantages of both the Grey Wolf Optimizer (GWO) and the Elephant Herding Optimisation (EHO), combining them for hybridization would be fantastic. Combining the GWO algorithm with the EHO algorithm increases its convergence speed, as well as its exploitation and exploration performances. Therefore, a new hybrid optimization approach is proposed in this study for selecting weights in SVM optimally. When compared to other machine learning methods, the suggested model is shown to be superior in its ability to handle such issues and to produce optimal solutions.

On one problem for a fourth-order mixed-type equation that degenerates inside and on the boundary of a domain

In the article, a nonlocal boundary value problem has been investigated for a fourth-order mixed-type equation degenerating inside and on the boundary of a domain. Applying the method of separation of variables to the problem under study, the spectral problem for an ordinary differential equation is obtained. The Green function of the last problem is constructed, with the help of which it is equivalently reduced to the Fredholm integral equation of the second kind with a symmetric kernel, which implies the existence of eigenvalues and the system of eigenfunctions for the spectral problem. The theorem of expansion of a given function into a uniformly convergent series with respect to the system of eigenfunctions is proved. Using the found integral equation and Mercer's theorem, a uniform convergence of some bilinear series depending on the found eigenfunctions is proved. The order of the Fourier coefficients is established. The solution of the problem under study is written as the sum of the Fourier series with respect to the system of eigenfunctions of the spectral problem. An estimate for the problem's solution is obtained, from which its continuous dependence on the given functions follows.

Anomaly detection for blueberry data using sparse autoencoder-support vector machine.

High-dimensional space includes many subspaces so that anomalies can be hidden in any of them, which leads to obvious difficulties in abnormality detection. Currently, most existing anomaly detection methods tend to measure distances between data points. Unfortunately, the distance between data points becomes more similar as the dimensionality of the input data increases, resulting in difficulties in differentiation between data points. As such, the high dimensionality of input data brings an obvious challenge for anomaly detection. To address this issue, this article proposes a hybrid method of combining a sparse autoencoder with a support vector machine. The principle is that by first using the proposed sparse autoencoder, the low-dimensional features of the input dataset can be captured, so as to reduce its dimensionality. Then, the support vector machine separates abnormal features from normal features in the captured low-dimensional feature space. To improve the precision of separation, a novel kernel is derived based on the Mercer theorem. Meanwhile, to prevent normal points from being mistakenly classified, the upper limit of the number of abnormal points is estimated by the Chebyshev theorem. Experiments on both the synthetic datasets and the UCI datasets show that the proposed method outperforms the state-of-the-art detection methods in the ability of anomaly detection. We find that the newly designed kernel can explore different sub-regions, which is able to better separate anomaly instances from the normal ones. Moreover, our results suggested that anomaly detection models suffer less negative effects from the complexity of data distribution in the space reconstructed by those layered features than in the original space.

On the speed of uniform convergence in Mercer's theorem

The classical Mercer's theorem claims that a continuous positive definite kernel K(x,y) on a compact set can be represented as ∑i=1∞λiϕi(x)ϕi(y) where {(λi,ϕi)} are eigenvalue-eigenvector pairs of the corresponding integral operator. This infinite representation is known to converge uniformly to the kernel K. We estimate the speed of this convergence in terms of the decay rate of eigenvalues and demonstrate that for 2m times differentiable kernels the first N terms of the series approximate K as O((∑i=N+1∞λi)mm+n) or O((∑i=N+1∞λi2)m2m+n). Finally, we demonstrate some applications of our results to a spectral characterization of integral operators with continuous roots and other powers.

Learning “best” kernels from data in Gaussian process regression. With application to aerodynamics

This paper introduces algorithms to select/design kernels in Gaussian process regression/kriging surrogate modeling techniques. We adopt the setting of kernel method solutions in ad hoc functional spaces, namely Reproducing Kernel Hilbert Spaces (RKHS), to solve the problem of approximating a regular target function given observations of it, i.e. supervised learning. A first class of algorithms is kernel flow, which was introduced in the context of classification in machine learning. It can be seen as a cross-validation procedure whereby a “best” kernel is selected such that the loss of accuracy incurred by removing some part of the dataset (typically half of it) is minimized. A second class of algorithms is called spectral kernel ridge regression, and aims at selecting a “best” kernel such that the norm of the function to be approximated is minimal in the associated RKHS. Within Mercer's theorem framework, we obtain an explicit construction of that “best” kernel in terms of the main features of the target function. Both approaches of learning kernels from data are illustrated by numerical examples on synthetic test functions, and on a classical test case in turbulence modeling validation for transonic flows about a two-dimensional airfoil.

On the solvability of nonlocal initial-boundary value problems for a partial differential equation of high even order

In the present paper, two non-local initial-boundary value problems have been formulated for a partial differential equation of high even order with a Bessel operator in a rectangular domain. The correctness of one of the considered problems has been investigated. To do this, applying the method of separation of variables to the problem under consideration, the spectral problem was obtained for an ordinary differential equation of high even order. The self-adjointness of the last problem was proved, which implies the existence of the system of its eigenfunctions, as well as orthonormality and completeness of this system. Further, the Green's function of the spectral problem was constructed, with the help of which it was equivalently reduced to the Fredholm integral equation of the second kind with symmetrical kernel. Using this integral equation and Mercer's theorem, the uniform convergence of some bilinear series depending on found eigenfunctions has been studied. The order of the Fourier coefficients was established. The solution of the considered problem has been written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and also the series obtained from it by term-by-term differentiation was proved. Using the method of spectral analysis, the uniqueness of the solution of the problem was proved. An estimate for the solution of the problem was obtained, from which its continuous dependence on the given functions follows.

A novel hybrid kernel function relevance vector machine for multi-task motor imagery EEG classification

Relevance vector machine (RVM) is a sparse Bayesian probability model commonly utilized in classification problems. Kernel functions are critical for the classification capacity of RVM. The kernel functions of RVM are not limited by the Mercer theorem, which differs RVM from the traditional support vector machine (SVM). As a typical local kernel function, the Gaussian kernel function has strong interpolating capacity, while the polynomial kernel function, a representative of global kernel function, is good at extrapolation. By combining the Gaussian kernel function and the polynomial kernel function, this paper proposed a novel hybrid kernel function RVM which is effective for classification at both local and global feature levels. Multi-task motor imagery electroencephalogram (EEG) classification is considered to validate the proposed method. Firstly, the phase space reconstruction (PSR) is employed to project EEG data from the time domain into the high-dimensional phase space, where the phase space common spatial pattern (PSCSP) features are extracted by using the “one versus one” common spatial pattern (OVO-CSP) strategy. Then the obtained PSCSP features are utilized as the input feature vectors of the proposed hybrid kernel function RVM for classification. The experimental results show that the proposed method improves the accuracy and Kappa coefficient for the multi-task motor imagery EEG classification problem. The main contributions of the paper include the novel hybrid kernel function RVM and the PSCSP features extracted from EEG.

Eigenvalues with respect to a weight for general boundary value problems on networks

In this work we analyze self-adjoint boundary value problems on networks for Schrödinger operators, in which a part of the boundary with a Neumann condition is always considered. We first characterize when the energy is positive semi-definite on the space of functions satisfying the null boundary conditions. To do this, the fundamental tools are the Doob transform and the discrete version of the trace function. Then, we raise eigenvalue problems with respect to a weight for general boundary value problems and we prove the discrete version of the Mercer Theorem. Finally, we apply the obtained results to a Dirichlet-Robin boundary value problem on a star-like network.

Weighted p-regular kernels for reproducing kernel Hilbert spaces and Mercer Theorem

Let [Formula: see text] be a finite measure space and consider a Banach function space [Formula: see text]. Motivated by some previous papers and current applications, we provide a general framework for representing reproducing kernel Hilbert spaces as subsets of Köthe–Bochner (vector-valued) function spaces. We analyze operator-valued kernels [Formula: see text] that define integration maps [Formula: see text] between Köthe–Bochner spaces of Hilbert-valued functions [Formula: see text] We show a reduction procedure which allows to find a factorization of the corresponding kernel operator through weighted Bochner spaces [Formula: see text] and [Formula: see text] — where [Formula: see text] — under the assumption of [Formula: see text]-concavity of [Formula: see text] Equivalently, a new kernel obtained by multiplying [Formula: see text] by scalar functions can be given in such a way that the kernel operator is defined from [Formula: see text] to [Formula: see text] in a natural way. As an application, we prove a new version of Mercer Theorem for matrix-valued weighted kernels.

Operators with Continuous Kernels

Let $\Omega \subset {\bf R}^d$ be open. We investigate conditions under which an operator $T$ on $L_2(\Omega)$ has a continuous kernel $K \in C(\overline \Omega \times \overline \Omega)$. In the centre of our interest is the condition $T L_2(\Omega) \subset C(\overline \Omega)$, which one knows for many semigroups generated by elliptic operators. This condition implies that $T^3$ has a kernel in $C(\overline \Omega \times \overline \Omega)$ if $T$ is self-adjoint and $\Omega$ is bounded, and the power $3$ is best possible. We also analyse Mercer's theorem in our context.

RETRACTED ARTICLE: Application of combined kernel function artificial intelligence algorithm in mobile communication network security authentication mechanism

Security threats, security requirements, and adopted security policies in the new mobile environment have changed. As a key issue in the development of mobile communication, the mobile communication network security authentication mechanism has attracted the attention of academic circles and industry, and research on its security has become a research hotspot. Firstly, aiming at the security threats and security requirements in the heterogeneous mobile communication network environment, the architecture of the security authentication mechanism in this mobile environment is proposed. The security architecture consists of three layers (transport layer, service layer, application layer) and four domains (user domain, access domain, network domain, and application domain), and defined security features. The security architecture includes functional entities such as public key infrastructure and application servers and defines new security features for security threats in the application domain and user domain, so that the security architecture can meet the heterogeneous mobile communication network environment. Security needs. Secondly, a linear combination kernel function is used to construct a new class of kernel functions with different characteristics in a linear combination and satisfy the Mercer theorem. The combined kernel function has both global kernel function and local kernel. The advantages function and their weight on the combined kernel function can be adjusted by the weight coefficient factor, and a better comprehensive prediction effect is obtained in the support vector machine model. Finally, the experimental results also show that the support vector regression machine model based on combined kernel function can achieve higher learning precision.

Domain Invariant Transfer Kernel Learning

Domain transfer learning generalizes a learning model across training data and testing data with different distributions. A general principle to tackle this problem is reducing the distribution difference between training data and testing data such that the generalization error can be bounded. Current methods typically model the sample distributions in input feature space, which depends on nonlinear feature mapping to embody the distribution discrepancy. However, this nonlinear feature space may not be optimal for the kernel-based learning machines. To this end, we propose a transfer kernel learning (TKL) approach to learn a domain-invariant kernel by directly matching source and target distributions in the reproducing kernel Hilbert space (RKHS). Specifically, we design a family of spectral kernels by extrapolating target eigensystem on source samples with Mercer's theorem. The spectral kernel minimizing the approximation error to the ground truth kernel is selected to construct domain-invariant kernel machines. Comprehensive experimental evidence on a large number of text categorization, image classification, and video event recognition datasets verifies the effectiveness and efficiency of the proposed TKL approach over several state-of-the-art methods.

Bimodules in crossed products of von Neumann algebras

In this paper, we study bimodules over a von Neumann algebra M in the context of an inclusion M⊆M⋊αG, where G is a discrete group acting on a factor M by outer ⁎-automorphisms. We characterize the M-bimodules X⊆M⋊αG that are closed in the Bures topology in terms of the subsets of G. We show that this characterization also holds for w⁎-closed bimodules when G has the approximation property (AP), a class of groups that includes all amenable and weakly amenable ones. As an application, we prove a version of Mercer's extension theorem for certain w⁎-continuous surjective isometric maps on X.

On Fourier–Haar coefficients of the function from the Marcinkiewicz space

The main objective of the paper is to describe asymptotic behaviour of Fourier–Haar coefficients of functions from Marcinkiewicz spaces. We also discuss the Cesàro summability and the almost convergence of sequences related to Fourier–Haar coefficients and generalise a result from [17]. Some analogues of Mercer's theorem for Fourier coefficients are proved.

Structure of feature spaces related to fuzzy similarity relations as kernels

Fuzzy similarity relations have been considered as kernels in the domain of machine learning in previous works. As a supplementary of this consideration, this paper focuses on construction of feature space related to fuzzy similarity relations as indefinite kernels. First we generalize the Mercer Theorem from positive definite kernels to fuzzy similarity relations. With this generalization we construct the feature space related to a fuzzy similarity relation as a particular Krein space. The proposed feature space is a naturally generalization of the existing Pseudo-Euclidean space from finite to infinite universe of discourses. At the end of this paper we develop an open problem to set up the connection between transitivity and the positive definiteness of a fuzzy similarity relation. Results in this paper complete the mathematical foundation for further applications of fuzzy similarity relations under the framework of kernel tricks.

Positive definiteness‎, ‎reproducing kernel Hilbert spaces and beyond

Positive definiteness‎, ‎reproducing kernel Hilbert spaces‎, ‎integral operators and Mercer's‎ ‎theorem in its various formats are common topics in many branches of‎ ‎mathematics. In this paper we review and upgrade upon some recent‎ ‎results that always involve at least two of them and indicate a few‎ ‎directions in which additional research could be carried out‎.

An extension of Mercer theorem to matrix-valued measurable kernels

We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space X into Cn. Given a finite measure μ on X, we represent the reproducing kernel K as a convergent series in terms of the eigenfunctions of a suitable compact operator depending on K and μ. Our result holds under the mild assumption that K is measurable and the associated Hilbert space is separable. Furthermore, we show that X has a natural second countable topology with respect to which the eigenfunctions are continuous and such that the series representing K uniformly converges to K on compact subsets of X×X, provided that the support of μ is X.

Stable Evaluation of Gaussian Radial Basis Function Interpolants

We provide a new way to compute and evaluate Gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter, i.e., for “flat” kernels. This work is motivated by the fundamental ideas proposed earlier by Bengt Fornberg and his coworkers. However, following Mercer's theorem, an $L_2(\mathbb{R}^d, \rho)$-orthonormal expansion of the Gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by Fornberg, Larsson, and Flyer and that is applicable in arbitrary space dimensions d. In addition to obtaining an accurate approximation of the radial basis function interpolant (using many terms in the series expansion of the kernel), we also propose and investigate a highly accurate least-squares approximation based on early truncation of the kernel expansion.

Approximation of Hopkins equation in Hilbert space and its application in polarized illumination modelling

In this article, we derive an analytical approximation to compute the aerial images on general geometric domains. Under the framework in Hilbert space, we develop a general approximation model for aerial image intensity, which is particularly valid for the computation of the aerial image with polarized illumination source. Our mathematical result is a generalization of classical Mercer's theorem provided that the integral kernel is Hilbert–Schmidt. By applying Synopsys's simulation tool Progen, we demonstrate the effectiveness of our proposed approximation with polarized illumination sources (65 nm and below). Since our approach setting is quite general, the proposed approximation in this article is suitable for modification in various applications.

Hidden space-based nonlinear discriminant feature extraction method

A novel nonlinear feature extraction method based on the scatter difference criterion in hidden space is developed. The main idea is that the original input space is first mapped into a hidden space through a hidden function, which is still referred to as the kernel function in the proposed method, and, in this space, feature extraction is conducted using the difference of between-class scatter and within-class scatter as the discriminant criterion. Different from the existing kernel-based feature extraction methods, the kernel functions used in the proposed method are not required to satisfy Mercer's theorem so that they can be chosen from a wide range. What is more important is that, due to the adoption of the scatter difference as the discriminant criterion for feature extraction, the proposed method essentially avoids the small sample size problem usually encountered in kernel Fisher discriminant analysis. Finally, extensive experiments have been performed on a subset of the FERET face database and the CENPARMI handwritten digital database. The experimental results indicate that the proposed method outperforms traditional scatter difference discriminant analysis in recognition performance.