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With view to applications to harmonic and stochastic analysis of infinite network/graph models, we introduce new tools for realizations and transforms of positive definite kernels (p.d.) \begin{document}$ K $\end{document} and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of factorizations: (i) Probabilistic: Starting with a positive definite kernel \begin{document}$ K $\end{document} we analyze associated Gaussian processes \begin{document}$ V $\end{document} . Properties of the Gaussian processes will be derived from certain factorizations of \begin{document}$ K $\end{document} , arising as a covariance kernel of \begin{document}$ V $\end{document} . (ii) Geometric analysis: We discuss families of measure spaces arising as boundaries for \begin{document}$ K $\end{document} . Our results entail an analysis of a partial order on families of p.d. kernels, a duality for operators and frames, optimization, Karhunen–Loeve expansions, and factorizations. Applications include a new boundary analysis for the Drury-Arveson kernel, and for certain fractals arising as iterated function systems; and an identification of optimal feature spaces in machine learning models.

The brain is able to calculate the distance and direction to the desired position based on grid cells. Extensive neurophysiological studies of rodent navigation have postulated the grid cells function as a metric for space, and have inspired many computational studies to develop innovative navigation approaches. Furthermore, grid cells may provide a general encoding scheme for high-order nonspatial information. Built upon existing neuroscience and machine learning work, this paper provides theoretical clarity on that the grid cell population codes can be taken as a metric for space. The metric is generated by a shift-invariant positive definite kernel via kernel distance method and embeds isometrically in a Euclidean space, and the inner product of the grid cell population code exponentially converges to the kernel. We also provide a method to learn the distribution of grid cell population efficiently. Grid cells, as a scalable position encoding method, can encode the spatial relationships of places and enable grid cells to outperform place cells in navigation. Further, we extend the grid cell to images encoding and find that grid cells embed images into a mental map, where geometric relationships are conceptual relationships of images. The theoretical model and analysis would contribute to establishing the grid cell code as a generic coding scheme for both spatial and conceptual spaces, and is promising for a multitude of problems across spatial cognition, machine learning and semantic cognition.

In this paper, we focus on the kernel-based solution of high-dimensional elliptic PDEs and propose an efficient algorithm to compute a trustable value of the shape parameter for a given positive definite kernel. The proposed strategy is based on the norm-minimal generalized interpolation problem coming from the optimality of radial basis function interpolation. Also, the effectiveness of the suggested algorithm is assessed by solving parabolic partial integro-differential equations arising in option pricing theory when the dynamic of the underlying asset is driven by a Levy process. The performance of the proposed algorithm is evaluated by some high-dimensional PDEs on regular and irregular computational domains. Indeed, the numerical results obtained with European put options under Merton’s model show the trustability of the suggested optimal shape parameter.

We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples.

Positive definite kernels and functions are considered. The key tool in the paper is the well-known main inequality for such kernels, namely, the Cauchy–Bunyakovskii inequality for the special inner product generated by a given positive definite kernel. It is shown that Ingham’s inequality (and, in particular, Hilbert’s inequality) is, essentially, the main inequality for the positive definite function $$\sin(\pi x)/x$$ on $$\mathbb{R}$$ and for a system of integer points. Using the main inequality, we prove new inequalities of Krein–Gorin type and Ingham’s inequality.

In this paper, we investigate conditionally completely positive definite kernels in the setting of Hilbert $$C^*$$ -modules. At first, we present a correspondence between conditionally completely positive definite kernels and completely positive definite kernels. Utilizing this relation, we give the Kolmogorov decomposition for a certain conditionally completely positive definite kernel. We present a characterization of conditionally completely positive definite kernels majorized by a kernel under some mild conditions. Finally, as an application of this decomposition, we find a sufficient condition for the existence of an extreme point of a convex set consisting of some special kernels.

The use of regional covariance descriptors to generate feature data represented by Symmetric Positive Definite (SPD) matrices from images or videos has become increasingly common in machine learning. However, SPD data itself does not constitute a vector space, and dictionary learning involves a large number of linear operations, so dictionary learning cannot be performed directly on SPD data. For this reason, a more common method is to map the SPD data to the Reproducing Kernel Hilbert Space (RKHS). The so-called kernel learning is to find the most suitable RKHS for specific tasks. RKHS can be uniquely generated by a kernel function. Therefore, RKHS learning can also be considered as kernel learning. In this article, there are two main contributions. The first contribution is to propose a framework which based on Kernel Learning and Riemannian Metric (KLRM). Usually the learnable kernel function framework is to learn some parameters in the kernel function. The second contribution is dictionary learning by applying KLRM to SPD data. The SPD data is transformed into the RKHS generated by KLRM, and RKHS after training provides the most suitable working space for dictionary learning. Under the proposed framework, we design a positive definite kernel function, which is defined by the Log-Euclidean metric. This function can be transformed into a corresponding Riemannian kernel. The experimental results provided in this paper is compared with other state-of-the-art algorithms for SPD data dictionary learning and show that the proposed algorithm achieves better results.

An iterative multistep kernel based method is proposed for the nonlinear Volterra integral equations and nonlinear Volterra integro-differential equations of fractional order, which can produce reliable globally smooth numerical solutions. An error estimate of the positive definite kernel interpolation and, the convergence and error analysis of the proposed iterative scheme are investigated. Here, we focused on positive definite radial basis kernels and further, a new and applicable shape parameter selection strategy is proposed. The proposed multi-step method set up and solve several small local problems instead of a single large problem which makes it suitable for problems with long-time simulations. In order to show the efficiency and versatility of the proposed method, some numerical experiments are reported. The comparison of the numerical results with the analytical solutions and the best-reported results in the literature confirm the good accuracy and applicability of the proposed method.

Time series clustering is an effective vehicle to explore and visualize the structure of a suite of time series. In this study, we generalize the kernel-based fuzzy c-means clustering algorithm by involving the dynamic time alignment kernel (DTAK) to cluster vector time series. In this method, the nonlinear time alignment embedded in DTAK makes the kernel-based fuzzy c-means available for sequences with variable lengths. However, it is noted that DTAK is not a strictly positive definite kernel, especially when the sample size is large. To overcome this, some strategies are presented to make the proposed algorithm available for large data sets. In addition, it is a challenge task to calculate the average sequence for a series of time series with different lengths. In kernel-based fuzzy c-means algorithm, it is not necessary to calculate the average sequence, which will increase the effectiveness of clustering techniques for time series. In the experiments, the kernel-based fuzzy c-means with DTAK is evaluated by both the data sets from the UCI KDD Archive and real-world data sets. Experimental results delivered by the proposed method demonstrate its effectiveness and robustness.

This article presents a comparative study of adaptive filter–based power curve models to estimate wind turbine power output. In the real world, wind turbines are never subjected to ideal conditions; thus, adaptive filter–based power curves serve best when estimating the power in a time-varying environment. Adaptive filter–based power curve is implemented using various algorithms like least mean square, kernel least mean square, recursive least square, and kernel recursive least square algorithms. All models have been developed on National Renewable Energy Laboratory datasets. The performance of various models has been compared on the basis of parameters like mean absolute error, root mean square error, and R-squared score. In addition to this, the learning curves of each method have been obtained to show the performance variation over time.

Indefinite kernels have attracted more and more attentions in machine learning due to its wider application scope than usual positive definite kernels. However, the research about indefinite kernel clustering is relatively scarce. Furthermore, existing clustering methods are mainly designed based on positive definite kernels which are incapable in indefinite kernel scenarios. In this paper, we propose a novel indefinite kernel clustering algorithm termed as indefinite kernel maximum margin clustering (IKMMC) based on the state-of-the-art maximum margin clustering (MMC) model. IKMMC tries to find a proxy positive definite kernel to approximate the original indefinite one and thus embeds a new F-norm regularizer in the objective function to measure the diversity of the two kernels, which can be further optimized by an iterative approach. Concretely, at each iteration, given a set of initial class labels, IKMMC firstly transforms the clustering problem into a classification one solved by indefinite kernel support vector machine (IKSVM) with an extra class balance constraint and then the obtained prediction labels will be used as the new input class labels at next iteration until the error rate of prediction is smaller than a pre-specified tolerance. Finally, IKMMC utilizes the prediction labels at the last iteration as the expected indices of clusters. Moreover, we further extend IKMMC from binary clustering problems to more complex multi-class scenarios. Experimental results have shown the superiority of our algorithms.

Let G be a locally compact abelian group and X be a nonempty compact set of G. Given a positive definite kernel $$K:G \times G \rightarrow {\mathbb {C}}$$ whose real part is continuous on the diagonal of $$G \times G$$ , we provide a necessary and sufficient condition for the linear span of the set $$\left\{ x \in X \mapsto K(x,y)\,:\, y \in X\right\} $$ to be dense in C(X) in the topology of uniform convergence.

The aim of the paper is to create a link between the theory of reproducing kernel Hilbert spaces (RKHS) and the notion of a unitary representation of a group or of a groupoid. More specifically, it is demonstrated on one hand, how to construct a positive definite kernel and an RKHS for a given unitary representation of a group(oid), and on the other hand how to retrieve the unitary representation of a group or a groupoid from a positive definite kernel defined on that group(oid) with the help of the Moore-Aronszajn theorem. The kernel constructed from the group(oid) representation is inspired by the kernel defined in terms of the convolution of functions on a locally compact group. Several illustrative examples of reproducing kernels related with unitary representations of groupoids are discussed in detail. The paper is concluded with the brief overview of the possible applications of the proposed constructions., 17 pages, 2 figures, 1 table

This paper explores the connection between fuzzy inference systems and kernel methods. Particularly, we explore the connection between the fuzzy-basis-function expansion of non-singleton fuzzy systems and regularized kernel methods with positive-definite kernels. We show that positive-definite kernel on fuzzy sets, i.e., a real-valued function defined on the set of fuzzy sets, is the core concept enabling such a connection. Furthermore, we rewrite the fuzzy-basis-function (FBF) expansion of non-singleton fuzzy systems in terms of a kernel-basis-function expansion with positive-definite kernels on fuzzy sets. The consequence of doing this is that fuzzy systems can be trained by regularized kernel methods, and the advantages of doing this are: 1) fuzzy systems can have more resistance to the curse of dimensionality, 2) they can explicitly regularize the function being approximated, 3) they can satisfy the Representer Theorem of kernel methods, which bounds the number of learned rules to the number of training observations, and, 4) they can learn functions in the functional space induced by the kernel on fuzzy sets. Consequently, from this perspective fuzzy systems can be regarded as kernel methods. Different from previous works, this connection relates the FBF-expansion directly to kernel methods without dropping the denominator of the expansion; also, thanks to the use of kernels on fuzzy sets, we provide an algorithm that makes it possible to drop the restriction of having fuzzy sets that share some of their antecedent parameters, e.g., Gaussian membership functions with the same variance in rule antecedents as is usually done in related works.

Cost volume is an essential component of recent deep models for optical flow estimation and is usually constructed by calculating the inner product between two feature vectors. However, the standard inner product in the commonly-used cost volume may limit the representation capacity of flow models because it neglects the correlation among different channel dimensions and weighs each dimension equally. To address this issue, we propose a learnable cost volume (LCV) using an elliptical inner product, which generalizes the standard inner product by a positive definite kernel matrix. To guarantee its positive definiteness, we perform spectral decomposition on the kernel matrix and re-parameterize it via the Cayley representation. The proposed LCV is a lightweight module and can be easily plugged into existing models to replace the vanilla cost volume. Experimental results show that the LCV module not only improves the accuracy of state-of-the-art models on standard benchmarks, but also promotes their robustness against illumination change, noises, and adversarial perturbations of the input signals.

In this article we study the field of Hilbertian metrics and positive definit (pd) kernels on probability measures, they have a real interest in kernel methods. Firstly we will make a study based on the Alpha-Beta-divergence to have a Hilbercan metric by proposing an improvement of this divergence by constructing it so that its is symmetrical the Alpha-Beta-Symmetric-divergence (ABS-divergence) and also do some studies on these properties but also propose the kernels associated with this divergence. Secondly we will do mumerical studies incorporating all proposed metrics/kernels into support vector machine (SVM). Finally we presented a algorithm for image classification by using our divergence.

We examine the third kind integral equations in Hölder class. The coefficients of the equations are piecewise strictly monotone functions having simple zeros. By singular integral equations theory, for solvability of considered equations, we give the necessary and sufficient conditions. Finding a solution is reduced to solving a regular integral equation of second kind. Keywords: Piecewise, Eigenvalues, Holomorphic, Singular operator

One class of singular integral equations of convolution type with Hilbert kernel is studied in the space $L^{2}[-\pi, \pi]$ in the article. Such equations can be changed into either a system of discrete equations or a discrete jump problem depending on some parameter via the discrete Laurent transform. We can thus solve the equations with an explicit representation of solutions under certain conditions.

In this paper, we study one class of generalized convolution-type singular integral equations in class {0}. Such equations are turned into complete singular integral equations with nodal points and further turned into boundary value problems for analytic function with discontinuous coefficients by Fourier transforms. For such equations, we will propose one method different from classical one and obtain the general solutions and their conditions of solvability in class {0}. Thus, this paper generalizes the theory of classical equations of convolution type.

We consider the systems of linear functional equations of the third kind with measurecompact operators in L2 and the general systems of linear integral equations of the third kind in L2. We propose a method for reducing these systems either to equivalent integral equations of the first kind with nuclear operators or to equivalent integral equations of the second kind with quasidegenerate Carleman kernels. Various exact and approximate methods for solving the arising integral equations are applicable.

In this paper, an effective numerical method to solve the Cauchy type singular Fredholm integral equations (CSFIEs) of the first kind is proposed. The collocation technique based on Bernstein polynomials is used for approximation the solution of various cases of CSFIEs. By transforming the problem into systems of linear algebraic equations, we see that this approach is computationally simple and attractive. Then the approximate solution of the problem in truncated series form is obtained by using the matrix form of this method. Convergence and error analyses of the presented method are mentioned. Finally, numerical experiments show the validity, accuracy, and efficiency of the proposed method.

This article presents the inverse of the kernel operator associated with the complete quadratic Lyapunov-Krasovskii functional for coupled differential-functional equations when the kernel operator is separable. Similar to the case of time-delay systems of retarded type, the inverse operator is instrumental in control synthesis. Unlike the power series expansion approach used in the previous literature, a direct algebraic method is used here. It is shown that the domain of definition of the infinitesimal generator is an invariant subspace of the inverse operator if it is an invariant subspace of the kernel operator. The process of control synthesis using the inverse operator is described, and a numerical example is presented using the sum-of-square formulation.

We consider the problem related to clustering of gamma-ray bursts (from "BATSE" catalogue) through kernel principal component analysis in which our proposed kernel outperforms results of other competent kernels in terms of clustering accuracy and we obtain three physically interpretable groups of gamma-ray bursts. The effectivity of the suggested kernel in combination with kernel principal component analysis in revealing natural clusters in noisy and nonlinear data while reducing the dimension of the data is also explored in two simulated data sets., Comment: 30 pages, 10 figures

In this paper, we present certain composition formulae of the pathway fractional integral operators associated with two extended Mittag-Leffler functions. Here, we find out the relevant connections of some particular cases of the main results with those earlier ones.

In this paper, the numerical technique based on Block Pulse functions (BPFs) has been developed to approximate the solutions of nonlinear VolterraFredholmHammerstein integral equations in two-dimensional spaces. These functions are orthogonal and have compact support on [0,1]. The proposed method reduces the integral equations to a system of nonlinear algebraic equations that can be easily solved by any numerical method. Also, the convergence of the proposed approach is discussed. Furthermore, in order to show the accuracy and reliability of the above-mentioned algorithm, the new approach is verified through some numerical examples. A numerical technique based on Block Pulse functions is proposed.Two-dimensional nonlinear VolterraFredholmHammerstein integral equations are firstly numerically solved.The proposed method is to transform the VolterraFredholm integral equations into a system of algebraic equations.The main characteristic behind the algorithm is operator theory.

By means of real analysis and weight functions, we obtain a few equivalent conditions of two kinds of Hardy-type integral inequalities with the non-homogeneous kernel and parameters. The constant factors related to the gamma function are proved to be the best possible. We also consider the operator expressions and some cases of homogeneous kernel.

In this paper we consider the approximation of noisy scattered data on the sphere by radial basis functions generated by a strictly positive definite kernel. The approximation is the minimizer in the native space for that kernel of a quadratic functional in which the smoothing term is a multiple of the square of the native space norm. The balance between data fitting and smoothness is controlled by a smoothing parameter, the choice of which should depend on the nature and magnitude of the noise. The main results concern the choice of that smoothing parameter, under the assumption that the noise is deterministic rather than random. Four strategies for choosing the smoothing parameter are considered: Morozov’s discrepancy principle, and three a priori strategies. For each of these strategies we derive an $$L_2$$ error bound. The error bounds are similar, with the discrepancy principle giving marginally the best bound. A numerical example supports the theoretical results.

We prove the existence theorem for solutions of geometrically nonlinear boundary-value problems for elastic shallow isotropic homogeneous shells with free edges under shear model of S. P. Timoshenko. Research method consists in the reduction of the original system of equilibrium equations to a single nonlinear equation for the components of transverse shear deformations. The basis of this method are integral representations for the generalized displacements, containing an arbitrary holomorphic functions, which are determined by the boundary conditions involving the theory of one-dimensional singular integral equations.

We consider linear functional equations of the third kind in L2 with arbitrary measurable coefficients and unbounded integral operators with kernels satisfying broad conditions. We propose methods for reducing these equations by linear continuous invertible transformations either to equivalent integral equations of the first kind with nuclear operators or to equivalent integral equations of the second kind with quasidegenerate Carleman kernels. To the integral equations obtained after the reduction, one can apply various exact and approximate methods of solution; in particular, the two approximate methods developed in this article.

We deal with several classes of integral transformations of the form f(x)↦D∫R+2(e−vcosh⁡(x−u)±e−vcosh⁡(x+u))f(u)h(v)dudv, where D is a special differential operator. In case D is the identical operator, we prove a Young type theorem for a generalized convolution operator related to the Fourier and the Kontorovich–Lebedev integral transforms. In case D is a certain finite order differential operator, we study the unitary property of these transforms and apply them to solve a class of integro-differential equations. Several classes of system of integral equations are also considered.

We study singular integral equations of convolution type with cosecant kernels and periodic coefficients in class L2[-π,π]. Such equations are transformed into a discrete jump problem or a discrete system of linear algebraic equations by using discrete Fourier transform. The conditions of Noethericity and the explicit solutions are obtained by means of the theory of classical boundary value problem and of the Fourier analysis theory. This paper will be of great significance for the study of improving and developing complex analysis, integral equations, and boundary value problems.

In this paper we introduce new conditions on a kernel of a linear Fredholm integral operator which turn out to be sufficient and necessary for that operator to map the space of functions of bounded variation in the sense of Jordan into itself. Furthermore, we apply those conditions to deal with the problem of the existence of solutions to nonlinear Hammerstein equations in that space. To achieve our goals we will use a topological degree approach as well as a fixed point approach.

View variation is one of the greatest challenges faced by the gait recognition research community. Recently, there are studies that model sets of gait features from multiple views as linear subspaces, which are known to form a special manifold called the Grassmann manifold. Conjecturing that modeling via linear subspace representation is not completely sufficient for gait recognition across view change, we take a step forward to consider non-linear subspace representation. A collection of multi-view gait features encapsulated in the form of a linear subspace is projected to the non-linear subspace through the expansion coefficients induced by kernel principal component analysis. Since subspace representation is inherently non-Euclidean, naive vectorization as input to the vector-based pattern analysis machines is expected to yield suboptimal accuracy performance. We deal with this difficulty by embedding the manifold in a Reproducing Kernel Hilbert Space (RKHS) through a positive definite kernel function defined on the Grassmann manifold. A closer examination reveals that the proposed approach can actually be interpreted as a doubly-kernel method. To be specific, the first kernel maps the linear subspace representation non-linearly to a feature space; while the second kernel permits the application of kernelization-enabled machines established for vector-based data on the manifold-valued multi-view gait features. Experiments on the CASIA gait database shows that the proposed doubly-kernel method is effective against view change in gait recognition.

Prototypical examples that best summarizes and compactly represents an underlying complex data distribution communicate meaningful insights to humans in domains where simple explanations are hard to extract. In this paper we present algorithms with strong theoretical guarantees to mine these data sets and select prototypes a.k.a. representatives that optimally describes them. Our work notably generalizes the recent work by Kim et al. (2016) where in addition to selecting prototypes, we also associate non-negative weights which are indicative of their importance. This extension provides a single coherent framework under which both prototypes and criticisms (i.e. outliers) can be found. Furthermore, our framework works for any symmetric positive definite kernel thus addressing one of the key open questions laid out in Kim et al. (2016). By establishing that our objective function enjoys a key property of that of weak submodularity, we present a fast ProtoDash algorithm and also derive approximation guarantees for the same. We demonstrate the efficacy of our method on diverse domains such as retail, digit recognition (MNIST) and on publicly available 40 health questionnaires obtained from the Center for Disease Control (CDC) website maintained by the US Dept. of Health. We validate the results quantitatively as well as qualitatively based on expert feedback and recently published scientific studies on public health, thus showcasing the power of our technique in providing actionability (for retail), utility (for MNIST) and insight (on CDC datasets) which arguably are the hallmarks of an effective data mining method., Accepted for publication in International Conference on Data Mining (ICDM) 2019

We are concerned with the polynomial stability and the integrability of the energy for second order integro-differential equations in Hilbert spaces with positive definite kernels, where the memory can be oscillating or sign-varying or not locally absolutely continuous (without any control conditions on the derivative of the kernel). For this stability problem, tools from the theory of existing positive definite kernels can not be applied. In order to solve the problem, we introduce and study a new mathematical concept – generalized positive definite kernel (GPDK). With the help of GPDK and its properties, we obtain an efficient criterion of the polynomial stability for evolution equations with such a general but more complicated and useful memory. Moreover, in contrast to existing positive definite kernels, GPDK allows us to directly express the decay rate of the related kernel.

We investigate the notion of conditionally positive definite in the context of Hilbert $$C^*$$ -modules and present a characterization of the conditionally positive definiteness in terms of the usual positive definiteness. We give a Kolmogorov type representation of conditionally positive definite kernels in Hilbert $$C^*$$ -modules. As a consequence, we show that a $$C^*$$ -metric space (S, d) is $$C^*$$ -isometric to a subset of a Hilbert $$C^*$$ -module if and only if $$K(s,t)=-d(s,t)^2$$ is a conditionally positive definite kernel. We also present a characterization of the order $$K'\le K$$ between conditionally positive definite kernels.

Volterra integrodifferential equations with unbounded operator coefficients in a Hilbert space that are operator models of integrodifferential equations arising in viscoelasticity theory are studied. These equations are shown to be well-posed in Sobolev spaces of vector functions, and spectral analysis is applied to the operator functions that are the symbols of the given equations.

We consider 3D singular integral equations that describe problems of interaction of an electromagnetic wave with 3D dielectric structures. By using the theory of singular integral equations, we reduce these equations to Fredholm integral equations of the second kind.

We determine a necessary and sufficient condition for the strict positive definiteness of a continuous and positive definite kernel on the torus.

In this paper, we try to study the numerical methods for solving integral equations from a new perspective-machine learning method. By means of the idea of kernel e -support vector regression machine ( e -SVR), we construct an optimization modeling for a class of Volterra-Fredholm integral equations and propose a novel numerical method for solving them. The proposed method has a certain versatility and can be used to solve some other kinds of integral equations. In order to verify the effectiveness of the proposed method, we perform a series of comparative experiments with six specific Volterra-Fredholm integral equations and a method proposed in Wang et?al. (2014). Experimental results show that the proposed method has a good approximation property.