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A fundamental drawback of kernel-based statistical models is their limited scalability to large data sets, which requires resorting to approximations. In this work, we focus on the popular Gaussian kernel and on techniques to linearize kernel-based models by means of random feature approximations. In particular, we do so by studying a less explored random feature approximation based on Maclaurin expansions and polynomial sketches. We show that such approaches yield poor results when modelling high-frequency data, and we propose a novel localization scheme that improves kernel approximations and downstream performance significantly in this regime. We demonstrate these gains on a number of experiments involving the application of Gaussian process regression to synthetic and real-world data of different data sizes and dimensions. [ABSTRACT FROM AUTHOR]

In the traditional active contour models, global region‐based methods fail to segment images with intensity inhomogeneity, and local region‐based methods are sensitive to initial contour. In this study, a novel fuzzy energy‐based active contour model is proposed to segment medical images, which are always corrupted by intensity inhomogeneity. In order to deal with intensity inhomogeneity, a local energy term is first constructed by substituting a non‐local weight for Gaussian kernel widely used in traditional local region‐based models. Second, the defined adaptive force can drive the level set function to adaptively increase or decrease according to image intensity information. Therefore, the initial contour can be initialised as a constant function, which eliminates the problem caused by contour initialisation. Moreover, the proposed active contour model is a convex function. Thus, the problem, resulting from optimising a non‐convex functional in the traditional active contour models, can be avoided. Experimental results validate the superiorities and effectiveness of the proposed model for image segmentation with comparisons of those yielded by several state‐of‐the‐art techniques. [ABSTRACT FROM AUTHOR]

Graph learning is an important approach for machine learning. Kernel method is efficient for constructing similarity graph. Single kernel isn’t sufficient for complex problems. In this paper we propose a framework for multi-kernel learning. We give a brief introduction of Gaussian kernel, LLE and sparse representation. Then we analyze the advantages and disadvantages of these methods and give the reason why the combine of these methods with random walk is efficient. We compare our method with baseline methods on real-world data sets. The results show the efficiency of our method. [ABSTRACT FROM AUTHOR]

We propose a Bayesian variable selection method in semi-parametric models with applications to genetic and epigenetic data (e.g., single nucleotide polymorphisms and DNA methylation, respectively). The data are individually standardized to reduce heterogeneity and facilitate simultaneous selection of categorical (single nucleotide polymorphisms) and continuous (DNA methylation) variables. The Gaussian reproducing kernel is applied to the transformed data to evaluate joint effect of the variables, which may include complex interactions between, e.g., single nucleotide polymorphisms and DNA methylation. Indicator variables are introduced to the model for the purpose of variable selection. The method is demonstrated and evaluated using simulations under different scenarios. We apply the method to identify informative DNA methylation sites and single nucleotide polymorphisms in a set of genes based on their joint effect on allergic sensitization. The selected single nucleotide polymorphisms and methylation sites have the potential to serve as early markers for allergy prediction, and consequently benefit medical and clinical research to prevent allergy before its manifestation. [ABSTRACT FROM AUTHOR]

A novel non-rigid object tracking based on interactive user-define marker and superpixel Gaussian kernel is proposed in this paper. In the initialization stage, instead of using the traditional bounding box to locate the targeted object, we have employed an interactive segmentation with user-defined marker to segment the object accurately in the first frame of the input video to avoid the background influence in the traditional bounding box. During the tracking stage, by using a Gaussian kernel as movement constraint, each superpixel is tracked independently to locate the object in the next frame. Experimental results show that the proposed method compared to state of the art methods can achieve better robustness and accuracy for various challenging video clips. [ABSTRACT FROM AUTHOR]

In this article, we provide decay rates for singular values of compact integral operators generated by power series kernels on either the unit sphere or the closed unit ball in ℝm + 1,m ≥ 1 under decay assumptions on the coefficients in the expansion of the kernel. The results are illustrated in concrete examples, including an integral operator generated by a Gaussian kernel on the sphere. [ABSTRACT FROM AUTHOR]

This paper introduces a kernel-based sampling approach for image reconstruction and meshing. Given an input image and a user-specified number of points, the proposed method can automatically generate adaptive distribution of samples globally. We formulate the problem as an optimization to reconstruct the image by summing a small number of Gaussian kernels to approximate the given target image intensity or density. Each Gaussian kernel has the fixed size and the same energy. After the optimization, the samples are well distributed according to the image intensity or density variations as well as faithfully preserved the feature edges, which can be used to reconstruct high-quality images. Finally, we generate the adaptive triangular or tetrahedral meshes based on the well-spaced samples in 2D and 3D images. Our results are compared qualitatively and quantitatively with the state-of-the-art in image sampling and reconstruction on several examples by using the standard measurement criteria. [ABSTRACT FROM AUTHOR]

This paper is devoted to two issues involved in the one-class support vector machine (OCSVM), i.e., the optimization algorithm and the kernel parameter selection. For appropriate choices of parameters, the primal maximum margin problem of OCSVM is equivalent to a nearest point problem. A generalized Gilbert (GG) algorithm is proposed to solve the nearest point problem. Compared with the algebraic algorithms developed for OCSVM, such as the well-known sequential minimal optimization (SMO) algorithm, the GG algorithm is a novel geometric algorithm that has an intuitive and explicit optimization target at each iteration. Moreover, an improved MIES (IMIES) is developed for the Gaussian kernel parameter selection. IMIES is implemented by constraining the geometric locations of edge and interior sample mappings relative to OCSVM separating hyper-planes. The experimental results on 2-D artificial datasets and benchmark datasets show that IMIES is able to select suitable kernel parameters, and the GG algorithm is computationally more efficient while achieving comparable accuracies to the SMO algorithm. [ABSTRACT FROM AUTHOR]

Considering blind image deconvolution as a statistical estimation problem, we propose an unbiased estimator of the prediction error – Mallows’ statistics C L – as a novel criterion for estimating a point spread function (PSF) from the degraded image only. The PSF is obtained by minimizing this new objective functional over a family of smoother filterings (with frequency-dependent regularization term). We then perform non-blind deconvolution using the popular BM3D algorithm. The C L -based framework is exemplified with a number of parametric PSF’s, involving a scaling factor that controls the blur size. A typical example of such parametrization is the Gaussian kernel. The experimental results show that the C L -minimization yields highly accurate estimates of the PSF parameters, which also result in a negligible loss of visual quality, compared to that obtained with the exact PSF. The highly competitive results demonstrate the great potential of developing more powerful blind deconvolution algorithms based on the C L -estimator. [ABSTRACT FROM AUTHOR]

In the circular restricted three-body problem (CR3BP), transit orbit is a class of orbit which can pass through the bottleneck region of the zero velocity curve and escapes from the vicinity of the primary or the secondary. This kind of orbit plays a very important role in the design of space exploration missions. A kind of low-energy interplanetary transfer, which is called Interplanetary Superhighway (IPS), can be realized by utilizing transit orbits. To use the transit orbit in actual mission design, a key issue is to find an algorithm which can separate the states corresponding to transit orbits from the states corresponding to other types of orbits rapidly. In fact, the distribution of transit orbit in the phase space has been investigated by numerical method, and a Fourier series approximation method has been introduced to describe the boundary of transit orbits. However, the Fourier series approximation method needs several hundred sets of Fourier series. The coefficients of these Fourier series are neither easy to be computed nor convenient to be stored, which makes the method can hardly be used in actual mission design. In this paper, the support vector machine (SVM) is used to classify the trajectories in the CR3BP. Using the Gaussian kernel, the 6-dimensional states in the CR3BP are mapped into an infinite-dimensional space, and the bound of the transit orbits is described by a hyperplane. A training data generation method is introduced, which reduces the size of training data by generating the states near the hyperplane. The numerical results show that the proposed algorithm gives the good correct rate of classification, and its computing speed is much faster than that of the Fourier series approximation method. [ABSTRACT FROM AUTHOR]

Object extraction from remote sensing images has long been an intensive research topic in the field of surveying and mapping. Most past methods are devoted to handling just one type of object, and little attention has been paid to improving the computational efficiency. In recent years, level set evolution (LSE) has been shown to be very promising for object extraction in the field of image processing because it can handle topological changes automatically while achieving high accuracy. However, the application of state-of-the-art LSEs is compromised by laborious parameter tuning and expensive computation. In this paper, we proposed two fast LSEs for man-made object extraction from high spatial resolution remote sensing images. We replaced the traditional mean curvature-based regularization term by a Gaussian kernel, and it is mathematically sound to do that. Thus, we can use a larger time step in the numerical scheme to expedite the proposed LSEs. Compared with existing methods, the proposed LSEs are significantly faster. Most importantly, they involve much fewer parameters while achieving better performance. Their advantages over other state-of-the-art approaches have been verified by a range of experiments. [ABSTRACT FROM PUBLISHER]

Most hyper-ellipsoidal clustering (HEC) approaches use the Mahalanobis distance as a distance metric. It has been proven that HEC, under this condition, cannot be realized since the cost function of partitional clustering is a constant. We demonstrate that HEC with a modified Gaussian kernel metric can be interpreted as a problem of finding condensed ellipsoidal clusters (with respect to the volumes and densities of the clusters) and propose a practical HEC algorithm that is able to efficiently handle clusters that are ellipsoidal in shape and that are of different size and density. We then try to refine the HEC algorithm by utilizing ellipsoids defined on the kernel feature space to deal with more complex-shaped clusters. The proposed methods lead to a significant improvement in the clustering results over K-means algorithm, fuzzy Cmeans algorithm, GMM-EM algorithm, and HEC algorithm based on minimum-volume ellipsoids using Mahalanobis distance. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a simple method to estimate the width of Gaussian kernel based on an adaptive scaling technique. The Gaussian kernel is widely employed in radial basis function (RBF) network, support vector machine (SVM), least squares support vector machine (LS-SVM), Kriging models, and so on. It is widely known that the width of the Gaussian kernel in these machine learning techniques plays an important role. Determination of the optimal width is a time-consuming task. Therefore, it is preferable to determine the width with a simple manner. In this paper, we first examine a simple estimate of the width proposed by Nakayama et al. Through the examination, four sufficient conditions for the simple estimate of the width are described. Then, a new simple estimate for the width is proposed. In order to obtain the proposed estimate of the width, all dimensions are equally scaled. A simple technique called the adaptive scaling technique is also developed. It is expected that the proposed simple method to estimate the width is applicable to wide range of machine learning techniques employing the Gaussian kernel. Through examples, the validity of the proposed simple method to estimate the width is examined. [Copyright &y& Elsevier]

Feature selection for classification in high-dimensional spaces can improve generalization, reduce classifier complexity, and identify important, discriminating feature "markers." For support vector machine (SVM) classification, a widely used technique is recursive feature elimination (RFE).We demonstrate that RFE is not consistent with margin maximization, central to the SVM learning approach. We thus propose explicit margin-based feature elimination (MFE) for SVMs and demonstrate both improved margin and improved generalization, compared with RFE. Moreover, for the case of a nonlinear kernel, we show that RFE assumes that the squared weight vector 2-norm is strictly decreasing as features are eliminated. We demonstrate this is not true for the Gaussian kernel and, consequently, RFE may give poor results in this case. MFE for nonlinear kernels gives better margin and generalization. We also present an extension which achieves further margin gains, by optimizing only two degrees of freedom--the hyperplane's intercept and its squared 2-norm--with the weight vector orientation fixed. We finally introduce an extension that allows margin slackness. We compare against several alternatives, including RFE and a linear programming method that embeds feature selection within the classifier design. On high-dimensional gene microarray data sets, University of California at Irvine (UCI) repository data sets, and Alzheimer's disease brain image data, MFE methods give promising results. [ABSTRACT FROM AUTHOR]

Abstract: Fractal dimension provides a measure of the complexity of a dynamic system, and contains the health information of a machine. The basics of regularization dimension and the effects of Gaussian kernel parameters on the regularization of a signal are introduced. Regularization dimension has advantages over other fractal dimensions because the scale-independent range can be selected according to the signal frequency components of interest. Experimental gearbox vibration signals are analyzed by means of spectral analysis firstly, and then according to the spectral structure, the scale-independent range is selected for computing the regularization dimension, which increases monotonically with increasing gear damage degree. Comparison with correlation dimension and kurtosis shows the advantages of regularization dimension in assessing the localized gear damage. [Copyright &y& Elsevier]

Abstract: Automated discrimination between walnut shell and pulp has become an imperative task in the walnut postharvest processing industry in the US. During the last several years, hyperspectral fluorescence imaging has been widely used to inspect agricultural products for quality and safety due to its full of spectral information and ability of identifying different chemical components in the subject. This paper studied the feasibility of analyzing the difference of black walnuts shell and pulp by hyperspectral fluorescence imaging. Meanwhile, a Gaussian-kernel based support vector machine (SVM) approach was used to classify the walnuts shell and pulp. Experiments result with an overall 90.3% recognition rate based on 6257 samples showed that hyperspectral fluorescence imaging and proposed classification method were effective in differentiation of walnuts shell and pulp. In order to further evaluate the performance of proposed SVM classifier, the experiment results of SVM were compared to that of traditional principal component analysis (PCA) and Fisher’s discriminant analysis (FDA) methods. The conclusion was made that SVM with the Gaussian kernel function method was better than PCA and FDA in classifying the walnuts shell and pulp. [Copyright &y& Elsevier]

Summary: Support vector machine (SVM) is one of the most elegant data mining engines developed most recently. It has been shown in various studies that SVM provides higher accuracy level than the local model in the chaotic time series analysis. Chaotic time series analysis usually requires a long data record and it is therefore computationally time consuming in addition to possible storage capacity problems. In this study a ridge linear regression is applied in a feature space. The feature space dimension of Gaussian kernel is infinite. With the use of a data sample set, the number of dimensions of feature space of Gaussian kernel can be estimated. The scheme can computationally be guaranteed to be faster and, at the same time, stable while the accuracy remains close to or much better than other existing techniques. Existing techniques used for comparisons are: (1) standard chaos technique; (2) Naïve; (3) ARIMA; (4) Inverse Approach; and (5) SVM coupled the decomposition method. The parameters involved are calibrated with an evolutionary algorithm, Shuffled Complex Evolution (SCE). The performance of the proposed method is tested on Tryggevælde catchment runoff and Mississippi river flow. Significantly higher prediction accuracies are obtained from the proposed scheme than from other existing techniques. In addition, the training speed of the scheme is very much faster than that of its counterparts (197 words<300 words). [Copyright &y& Elsevier]

We consider the problem of Tikhonov regularization with a general convex loss function: this formalism includes support vector machines and regularized least squares. For a family of kernels that includes the Gaussian, parameterized by a "bandwidth" parameter σ, we characterize the limiting solution as σ → ∞. In particular, we show that if we set the regularization parameter λ = ̃λσ-2p, the regularization term of the Tikhonov problem tends to an indicator function on polynomials of degree ⌊p⌋ (with residual regularization in the case where p ∈ 핫). The proof rests on two key ideas: epi-convergence, a notion of functional convergence under which limits of minimizers converge to minimizers of limits, and a value-based formulation of learning, where we work with regularization on the function output values (y) as opposed to the function expansion coefficients in the RKHS. Our result generalizes and unifies previous results in this area. [ABSTRACT FROM AUTHOR]

We present a novel clustering method using the approach of support vector machines. Data points are mapped by means of a Gaussian kernel to a high dimensional feature space, where we search for the minimal enclosing sphere. This sphere, when mapped back to data space, can separate into several components, each enclosing a separate cluster of points. We present a simple algorithm for identifying these clusters. The width of the Gaussian kernel controls the scale at which the data is probed while the soft margin constant helps coping with outliers and overlapping clusters. The structure of a dataset is explored by varying the two parameters, maintaining a minimal number of support vectors to assure smooth cluster boundaries. We demonstrate the performance of our algorithm on several datasets. [ABSTRACT FROM AUTHOR]

A set-valued information system (SIS) is the generalization of a single-valued information system. This article explores uncertainty measurement for a SIS by using Gaussian kernel. The fuzzy T c o s -equivalence relation lead by a SIS is first obtained by using Gaussian kernel. Then, information structures in this SIS are described by set vectors. Next, dependence between information structures is presented and properties of information structures are investigated. Lastly, uncertainty measures of a SIS are presented by using its information structures. Moreover, effectiveness analysis is done to assess the feasibility of our presented measures. The consequence of this article will help us understand the intrinsic properties of uncertainty in a SIS. [ABSTRACT FROM AUTHOR]