Proposed Kernel Functions Research Articles

Contrastive Multi-View Kernel Learning

Kernel method is a proven technique in multi-view learning. It implicitly defines a Hilbert space where samples can be linearly separated. Most kernel-based multi-view learning algorithms compute a kernel function aggregating and compressing the views into a single kernel. However, existing approaches compute the kernels independently for each view. This ignores complementary information across views and thus may result in a bad kernel choice. In contrast, we propose the Contrastive Multi-view Kernel - a novel kernel function based on the emerging contrastive learning framework. The Contrastive Multi-view Kernel implicitly embeds the views into a joint semantic space where all of them resemble each other while promoting to learn diverse views. We validate the method's effectiveness in a large empirical study. It is worth noting that the proposed kernel functions share the types and parameters with traditional ones, making them fully compatible with existing kernel theory and application. On this basis, we also propose a contrastive multi-view clustering framework and instantiate it with multiple kernel k-means, achieving a promising performance. To the best of our knowledge, this is the first attempt to explore kernel generation in multi-view setting and the first approach to use contrastive learning for a multi-view kernel learning.

A New full-newton step infeasible interior-point method for P∗(κ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_*(\\kappa )$$\\end{document}-linear Complementarity problem

In this paper, we suggest a new infeasible interior-point method (IIPM) for P∗(κ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{*}(\\kappa )$$\\end{document}-linear complementarity problem (P∗(κ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{*}(\\kappa )$$\\end{document}-LCP) based on a new class of kernel functions with full-Newton steps. Each iteration of the algorithm consists of a feasible step and a few centering steps. The feasible step is defined by the newly proposed kernel functions, while the centering step is determined by using Newton’s method. New class of kernel functions includes a logarithmic kernel function in Roos (SIAM J Optim 16:1110–1136, 2006) and a trigonometric kernel function in Moslemi and Kheirfam (Optim Lett 13:127–157, 2019), Kheirfam and Haghighi (Commun Comb Optim 3:51–70, 2018), Fathi-Hafshejani et al. (J Appl Math Comput 48:111–128, 2015) as special cases. And we show that the proposed algorithm has the best known complexity for P∗(κ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_*(\\kappa )$$\\end{document}-LCP for such a method and present some numerical results.

Generalized Legendre Polynomials for Support Vector Machines (SVMS) Classification

In this paper, we introduce a set of new kernel functions derived from the generalized Legendre polynomials to obtain more robust and higher support vector machine (SVM) classification accuracy. The generalized Legendre kernel functions are suggested to provide a value of how two given vectors are like each other by changing the inner product of these two vectors into a greater dimensional space. The proposed kernel functions satisfy the Mercer’s condition and orthogonality properties for reaching the optimal result with low number support vector (SV). For that, the new set of Legendre kernel functions could be utilized in classification applications as effective substitutes to those generally used like Gaussian, Polynomial and Wavelet kernel functions. The suggested kernel functions are calculated in compared to the current kernels such as Gaussian, Polynomial, Wavelets and Chebyshev kernels by application to various non-separable data sets with some attributes. It is seen that the suggested kernel functions could give competitive classification outcomes in comparison with other kernel functions. Thus, on the basis test outcomes, we show that the suggested kernel functions are more robust about the kernel parameter change and reach the minimal SV number for classification generally.

Support vector machines for classification of low birth weight in Indonesia

This paper proposes support vector machines (SVMs), which is currently one of the most popular algorithms in machine learning (ML), in order to classify the low birth weight (LBW) data. The main objectives of this study are to predict the classification of LBW data in Indonesia based on the SVMs andto compare the performance of the proposed SVMs with the binary logistic regression as the most common model for classification of LBW data. The obtained samples were based on the results of Indonesian Demographic and Health Survey in 2012. The results showed that SVMs with four kernel functions (linear, radial, polynomial and hyperbolic tangent) were fit well to the LBW data in Indonesia. Furthermore, the constructed SVMs based on linear kernel function had the best performance among the SVMs with the other proposed kernel functions. This research also concluded that the SVMs based on linear kernel competed well with thebinary logistic regression forclassification LBW data in Indonesia.

GENERALIZED LEGENDRE POLYNOMIALS FOR SUPPORT VECTOR MACHINES (SVMS) CLASSIFICATION

In this paper, we introduce a set of new kernel functions derived from the generalized Legendre polynomials to obtain more robust and higher support vector machine (SVM) classification accuracy. The generalized Legendre kernel functions are suggested to provide a value of how two given vectors are like each other by changing the inner product of these two vectors into a greater dimensional space. The proposed kernel functions satisfy the Mercer’s condition and orthogonality properties for reaching the optimal result with low number support vector (SV). For that, the new set of Legendre kernel functions could be utilized in classification applications as effective substitutes to those generally used like Gaussian, Polynomial and Wavelet kernel functions. The suggested kernel functions are calculated in compared to the current kernels such as Gaussian, Polynomial, Wavelets and Chebyshev kernels by application to various non-separable data sets with some attributes. It is seen that the suggested kernel functions could give competitive classification outcomes in comparison with other kernel functions. Thus, on the basis test outcomes, we show that the suggested kernel functions are more robust about the kernel parameter change and reach the minimal SV number for classification generally.

Primal–dual interior-point method for linear optimization based on a kernel function with trigonometric growth term

In this paper, we propose a large-update primal–dual interior-point algorithm for linear optimization problems based on a new kernel function with a trigonometric growth term. By simple analysis, we prove that in the large neighbourhood of the central path, the worst case iteration complexity of the new algorithm is bounded above by , which matches the currently best known iteration bound for large-update methods. Moreover, we show that, most of the so far proposed kernel functions can be rewritten as a kernel function with trigonometric growth term. Finally, numerical experiments on some test problems confirm that the new kernel function is well promising in practice in comparison with some existing kernel functions in the literature.

Improving efficiency of SPH method for simulation of free surface flows using a new treatment of Neumann boundary conditions

In this study, simplified SPH formulations are introduced using straightforward mathematical operations for simulation of incompressible inviscid free surface flows. The governing equations of such flows including the mass and momentum conservations are solved in a Lagrangian form using a pressure projection method. The unknown pressure field at each time step is obtained by solving a pressure Poisson equation enforcing incompressibility constraint. Free surface location during simulation time can be predicted by the particles whose density is below a set point. Wall boundaries are also represented by particles whose positions are fixed. One of the simplest methods for modeling wall boundary conditions is solving the pressure Poisson equation on the wall particles to prevent the inner fluid particles from accumulating in vicinity of the walls. Several lines of dummy particles are also required to be placed in outer region of the walls to satisfy the Neumann pressure boundary conditions by setting the pressure of a dummy particle to that of a wall particle in the normal direction of the solid walls. Initial arrangement of these particles can be uneasy task especially for non-geometric computational domains. Two different strategy are proposed and used here to obviate the requirement of placing the dummy particles. In the first strategy, the contribution of dummy particles is indirectly included in the numerical computations based on their fixed positions. In the second and main strategy, dummy particles are completely eliminated from the computations and the wall conditions can be modeled only with the wall particles. The efficiency and accuracy of both strategies in modeling the wall conditions are examined in simulation of dam breaking flow using two proposed kernel functions and the well-known Spline kernel function. The numerical results show that significant reduction in computational time can be achieved by employing one of the proposed kernel functions and the other proposed kernel function works well with the second strategy in modeling wall conditions.

Two-class support vector machine with new kernel function based on paths of features for predicting chemical activity

Information and computer science fields such as machine learning and graph theory are implemented in chemoinformatics to discover the properties of chemical compounds. This paper presents a new algorithm based on the two-class support vector machine (SVM) model, which has new kernel functions for paths of features, enabling the prediction of chemical compound activity. Initially, we extract all paths of features (star subgraphs) with certain lengths, and we encode them depending on their structure in the graphs. Then, we use these codes to construct two relationship matrices between those paths. These matrices contain common and different sub-paths between paths of stars. The number of sub-paths/paths for each compound is passed to the proposed kernel functions in the two-class SVM to predict the activity of chemical compounds. The relationship matrices created by the proposed algorithm help to reduce the number of features, which improves prediction accuracy. We apply the proposed algorithm with and without feature selection using two benchmark datasets, specifically, the monoamine oxidase (MAO) dataset and the AIDS antiviral screen database of active compound dataset, which have 68 and 2000 chemical compounds, respectively. We perform comparative experiments for the proposed kernel functions and many other two-class SVM prediction methods, and the results before feature selection show prediction accuracies of 94% and 99.5% for MAO and AIDS, respectively. After selection, the prediction accuracies are 96% and 99.5% for MAO and AIDS, respectively.

New parameterized kernel functions for linear optimization

Recent studies on the kernel function-based primal-dual interior-point algorithms indicate that a kernel function not only represents a measure of the distance between the iteration and the central path, but also plays a critical role in improving the computational complexity of an interior-point algorithm. In this paper, we propose a new class of parameterized kernel functions for the development of primal-dual interior-point algorithms for solving linear programming problems. The properties of the proposed kernel functions and corresponding parameters are investigated. The results lead to a complexity bounds of $${O\left(\sqrt{n}\,{\rm log}\,n\,{\rm log}\,\frac{n}{\epsilon}\right)}$$ for the large-update primal-dual interior point methods. To the best of our knowledge, this is the best known bound achieved.

Edit distance-based kernel functions for structural pattern classification

A common approach in structural pattern classification is to define a dissimilarity measure on patterns and apply a distance-based nearest-neighbor classifier. In this paper, we introduce an alternative method for classification using kernel functions based on edit distance. The proposed approach is applicable to both string and graph representations of patterns. By means of the kernel functions introduced in this paper, string and graph classification can be performed in an implicit vector space using powerful statistical algorithms. The validity of the kernel method cannot be established for edit distance in general. However, by evaluating theoretical criteria we show that the kernel functions are nevertheless suitable for classification, and experiments on various string and graph datasets clearly demonstrate that nearest-neighbor classifiers can be outperformed by support vector machines using the proposed kernel functions.

Singular meshless method using double layer potentials for exterior acoustics

Time-harmonic exterior acoustic problems are solved by using a singular meshless method in this paper. It is well known that the source points cannot be located on the real boundary, when the method of fundamental solutions (MFS) is used due to the singularity of the adopted kernel functions. Hence, if the source points are right on the boundary the diagonal terms of the influence matrices cannot be derived. Herein we present an approach to obtain the diagonal terms of the influence matrices of the MFS for the numerical treatment of exterior acoustics. By using the regularization technique to regularize the singularity and hypersingularity of the proposed kernel functions, the source points can be located on the real boundary and therefore the diagonal terms of influence matrices are determined. We also maintain the prominent features of the MFS, that it is free from mesh, singularity, and numerical integration. The normal derivative of the fundamental solution of the Helmholtz equation is composed of a two-point function, which is one of the radial basis functions. The solution of the problem is expressed in terms of a double-layer potential representation on the physical boundary based on the potential theory. The solutions of three selected examples are used to compare with the results of the exact solution, conventional MFS, boundary element method, and Dirichlet-to-Neumann finite element method. Good numerical performance is demonstrated by close agreement with other solutions.

Improved calculation of field variables in the domain based on BEM

BEM-based formulations for calculating field variables at internal points are well established and are effective, especially when only certain regions of the domain are of interest. Numerical integration problems exist while evaluating interior displacements and stresses near the boundary.The aim of this paper is to derive kernel functions, which allow the calculation of displacements and stresses in the frequency domain of elastodynamics. An easy way to improve the accuracy of the calculation of displacements in the boundary layer is also presented.The derived functions approximate the static solution, when the frequency tends to zero. The results obtained with the new kernel functions are compared with analytical 1-D and 3-D solutions of vibrating rods.The analytical solution is based on the Pochhammer equation for cylindrical bars adapted to given boundary conditions.The results presented indicate the effectiveness of the proposed kernel functions.