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In this paper, we present an algorithm for clustering multidimensional data, which we named TreeKDE. It is based on a tree structure decision associated with the optimization of the one-dimensional kernel density estimator function constructed from the orthogonal projections of the data on the coordinate axes. Among the main features of the proposed algorithm, we highlight the automatic determination of the number of clusters and their insertion in a rectangular region. Comparative numerical experiments are presented to illustrate the performance of the proposed algorithm and the results indicate that the TreeKDE is efficient and competitive when compared to other algorithms from the literature. Features such as simplicity and efficiency make the proposed algorithm an attractive and promising research field, which can be used as a basis for its improvement, and also for the development of new clustering algorithms based on the association between decision tree and kernel density estimator. [ABSTRACT FROM AUTHOR]

We propose a kernel-based partial permutation test for checking the equality of functional relationship between response and covariates among different groups. The main idea, which is intuitive and easy to implement, is to keep the projections of the response vector Y on leading principle components of a kernel matrix fixed and permute Y's projections on the remaining principle components. The proposed test allows for different choices of kernels, corresponding to different classes of functions under the null hypothesis. First, using linear or polynomial kernels, our partial permutation tests are exactly valid in finite samples for linear or polynomial regression models with Gaussian noise; similar results straightforwardly extend to kernels with finite feature spaces. Second, by allowing the kernel feature space to diverge with the sample size, the test can be large-sample valid for a wider class of functions. Third, for general kernels with possibly infinite-dimensional feature space, the partial permutation test is exactly valid when the covariates are exactly balanced across all groups, or asymptotically valid when the underlying function follows certain regularized Gaussian processes. We further suggest test statistics using likelihood ratio between two (nested) Gaussian process regression models, and propose computationally efficient algorithms utilizing the EM algorithm and Newton's method, where the latter also involves Fisher scoring and quadratic programming and is particularly useful when EM suffers from slow convergence. Extensions to correlated and non-Gaussian noises have also been investigated theoretically or numerically. Furthermore, the test can be extended to use multiple kernels together and can thus enjoy properties from each kernel. Both simulation study and application illustrate the properties of the proposed test. [ABSTRACT FROM AUTHOR]

In this article, we propose a novel semi-analytical method to model complicated material interfaces by using a Gaussian kernel function and the resulting error function (erf) available in the symbolic toolbox of MATLAB. The proposed method provides an analytical expression for the smoothened material properties via a diffused representation of material interfaces that eliminates the requirement to track material interfaces explicitly while implementing the finite element method. In this article, we have demonstrated the proposed diffused representation approach for the homogenization of composites with periodic microstructures. We have studied the influence of varying shapes and sizes of inclusion on the overall effective material properties and the elastic response of the homogenized composites. However, the method is very generic and can be applied to obtain an analytical expression for smoothened material properties for any complicated material interface that may be useful to design high-performance topologically interlocked composites. [ABSTRACT FROM AUTHOR]

The most commonly used kernel function of support vector machine (SVM) in nonlinear separable dataset in machine learning is Gaussian kernel, also known as radial basis function. The Gaussian kernel decays exponentially in the input feature space and uniformly in all directions around the support vector, causing hyper-spherical contours of kernel function. In this study, an adaptive kernel function is designed based on the Gaussian kernel, which is used in SVM. While the sigma parameter is determined as an arbitrary value in the traditional Gaussian kernel, a modified Gaussian kernel method is used that calculates an adaptive value depending on the input vectors in the proposed kernel function. The proposed kernel function is compared with the linear, polynomial and Gaussian kernels commonly used in support vector machines. The results show that the proposed kernel function performs well on separable linear and nonlinear datasets compared to other kernel functions. It is also compared to state-of-the-art support vector machine kernels. [ABSTRACT FROM AUTHOR]

Predicting the power obtained at the output of the photovoltaic (PV) system is fundamental for the optimum use of the PV system. However, it varies at different times of the day depending on intermittent and nonlinear environmental conditions including solar irradiation, temperature and the wind speed, Shortterm power prediction is vital in PV systems to reconcile generation and demand in terms of the cost and capacity of the reserve. In this study, a Gaussian kernel based Support Vector Regression (SVR) prediction model using multiple input variables is proposed for estimating the maximum power obtained from using perturb observation method in the different irradiation and the different temperatures for a short-term in the DC-DC boost converter at the PV system. The performance of the kernel-based prediction model depends on the availability of a suitable kernel function that matches the learning objective, since an unsuitable kernel function or hyper parameter tuning results in significantly poor performance. In this study for the first time in the literature both maximum power is obtained at maximum power point and short-term maximum power estimation is made. While evaluating the performance of the suggested model, the PV power data simulated at variable irradiations and variable temperatures for one day in the PV system simulated in MATLAB were used. The maximum power obtained from the simulated system at maximum irradiance was 852.6 W. The accuracy and the performance evaluation of suggested forecasting model were identified utilizing the computing error statistics such as root mean square error (RMSE) and mean square error (MSE) values. MSE and RMSE rates which obtained were 4.5566 * 10-04 and 0.0213 using ANN model. MSE and RMSE rates which obtained were 13.0000 * 10-04 and 0.0362 using SWD-FFNN model. Using SVR model, 1.1548 * 10-05 MSE and 0.0034 RMSE rates were obtained. In the short-term maximum power prediction, SVR gave higher prediction performance according to ANN and SWD-FFNN. [ABSTRACT FROM AUTHOR]

Bayesian methods were studied in this paper using deep neural networks. We are interested in variational autoencoders, where an encoder approaches the true posterior and the decoder approaches the direct probability. Specifically, we applied these autoencoders for unsteady and compressible fluid flows in aircraft engines. We used inferential methods to compute a sharp approximation of the posterior probability of these parameters with the transient dynamics of the training velocity fields and to generate plausible velocity fields. An important application is the initialization of transient numerical simulations of unsteady fluid flows and large eddy simulations in fluid dynamics. It is known by the Bayes theorem that the choice of the prior distribution is very important for the computation of the posterior probability, proportional to the product of likelihood with the prior probability. Hence, we propose a new inference model based on a new prior defined by the density estimate with the realizations of the kernel proper orthogonal decomposition coefficients of the available training data. We numerically show that this inference model improves the results obtained with the usual standard normal prior distribution. This inference model was constructed using a new algorithm improving the convergence of the parametric optimization of the encoder probability distribution that approaches the posterior. This latter probability distribution is datatargeted, similarly to the prior distribution. This new generative approach can also be seen as an improvement of the kernel proper orthogonal decomposition method, for which we do not usually have a robust technique for expressing the pre-image in the input physical space of the stochastic reduced field in the feature high-dimensional space with a kernel inner product. [ABSTRACT FROM AUTHOR]

In data processing, measurement of uncertainty is one of the significant evaluation tools, which can describe the uncertainty essence of data. So far, there are few measurable tools to study the uncertainty of type-2 fuzzy information systems (TFISs-2) (the expanded models of fuzzy information systems). This paper is devoted to looking for effective indicators to describe the uncertainty of TFISs-2. The fuzzy T cos -similarity relations are first introduced, which are generated by TFISs-2 based on Gaussian kernel. Then, the fuzzy information structures are defined on account of this fuzzy T cos -similarity relation. Next, two measures constructed from the upper and lower approximations are given for TFISs-2, that is, δ -accuracy and δ -roughness, which are used to reflect the degree of accuracy and inaccuracy of information by numerical form. Furthermore, by combining roughness and entropy, the δ -rough entropy is investigated. Finally, the practicability of proposed measures is tested by a numerical experiment. The experimental results show that the δ -rough entropy is efficacious and applicable for TFISs-2. [ABSTRACT FROM AUTHOR]

Out-of-focus blur is a common image degradation phenomenon that occurs in case of lens defocusing. The out-of-focus blur kernel is usually modeled as a Gaussian function or a uniform disk in previous work. In this paper, we propose that it can be more accurately depicted using the generalized Gaussian (GG) function. This is motivated by the theoretical analysis of the out-of-focus blur and the practical observation of real blur kernels. We show that as the out-of-focus blur kernels are of specific shapes, the GG function can be further simplified to a single-parameter model. We estimate the parameter of the GG blur kernel from image patches containing step edges, and obtain the clear image by non-blind image deblurring. Experimental results validate that the proposed GG blur kernel estimation algorithm outperforms the state-of-the-art ones deploying either parametric (disk and Gaussian) or nonparametric kernels, and consequently benefits the image deblurring process. [ABSTRACT FROM AUTHOR]

Support vector machine (SVM) is one of the well-known learning algorithms for classification and regression problems. SVM parameters such as kernel parameters and penalty parameter have a great influence on the complexity and performance of predicting models. Hence, the model selection in SVM involves the penalty parameter and kernel parameters. However, these parameters are usually selected and used as a black box, without understanding the internal details. In this paper, the behavior of the SVM classifier is analyzed when these parameters take different values. This analysis consists of illustrative examples, visualization, and mathematical and geometrical interpretations with the aim of providing the basics of kernel functions with SVM and to show how it works to serve as a comprehensive source for researchers who are interested in this field. This paper starts by highlighting the definition and underlying principles of SVM in details. Moreover, different kernel functions are introduced and the impact of each parameter in these kernel functions is explained from different perspectives. [ABSTRACT FROM AUTHOR]

Summary: Surrogate modeling techniques have been increasingly developed for optimization and uncertainty quantification problems in many engineering fields. The development of surrogates requires modeling high‐dimensional and nonsmooth functions with limited information. To this end, the hybrid surrogate modeling method, where different surrogate models are combined, offers an effective solution. In this paper, a new hybrid modeling technique is proposed by combining polynomial chaos expansion and kernel function in a sparse Bayesian learning framework. The proposed hybrid model possesses both the global characteristic advantage of polynomial chaos expansion and the local characteristic advantage of the Gaussian kernel. The parameterized priors are utilized to encourage the sparsity of the model. Moreover, an optimization algorithm aiming at maximizing Bayesian evidence is proposed for parameter optimization. To assess the performance of the proposed method, a detailed comparison is made with the well‐established PC‐Kriging technique. The results show that the proposed method is superior in terms of accuracy and robustness. [ABSTRACT FROM AUTHOR]

Abstract This paper discusses a method to estimate the expected value of the Gaussian kernel in the presence of incomplete data. We show how, under the general assumption of a missing-at-random mechanism, the expected value of the Gaussian kernel function has a simple closed-form solution. Such a solution depends only on the parameters of the Gamma distribution which is assumed to represent squared distances. Furthermore, we show how the parameters governing the Gamma distribution depend only on the non-central moments of the kernel arguments, via the second-order moments of their squared distance, and can be estimated by making use of any parametric density estimation model of the data distribution. We approximate the data distribution with the maximum likelihood estimate of a Gaussian mixture distribution. The validity of the method is empirically assessed, under a range of conditions, on synthetic and real problems and the results compared to existing methods. For comparison, we consider methods that indirectly estimate a Gaussian kernel function by either estimating squared distances or by imputing missing values and then computing distances. Based on the experimental results, the proposed method consistently proves itself an accurate technique that further extends the use of Gaussian kernels with incomplete data. Highlights • We propose a method to estimate the expected value of a Gaussian kernel in the presence of incomplete data. • Our method deals with missing data directly and do not rely on any imputation method. • The proposed formulation models the distribution of the data using a Gaussian Mixture Model (GMM). • The validity of our method is empirically assessed under a range of conditions on simulated and real problems. • Results showed that our method was able to better estimate Gaussian kernels when compared to other missing data treatments. [ABSTRACT FROM AUTHOR]

A novel mesh-free heuristic method for solving differential equations is proposed. The new approach can cope with linear, nonlinear, and partial differential equations (DE), and systems of DEs. Candidate solutions are expressed using a linear combination of kernel functions. Thus, the original problem is transformed into an optimization problem that consists in finding the parameters that define each kernel. The new optimization problem is solved applying a Covariance Matrix Adaptation Evolution Strategy. To increase the accuracy of the results, a Downhill Simplex local search is applied to the best solution found by the mentioned evolutionary algorithm. Our method is applied to 32 differential equations extracted from the literature. All problems are successfully solved, achieving competitive accuracy levels when compared to other heuristic methods. A simple comparison with numerical methods is performed using two partial differential equations to show the pros and cons of the proposed algorithm. To verify the potential of this approach with a more practical problem, an electric circuit is analyzed in depth. The method can obtain the dynamic behavior of the circuit in a parametric way, taking into account different component values. [ABSTRACT FROM AUTHOR]

The purpose of kernel adaptive filtering (KAF) is to map input samples into reproducing kernel Hilbert spaces and use the stochastic gradient approximation to address learning problems. However, the growth of the weighted networks for KAF based on existing kernel functions leads to high computational complexity. This paper introduces a reduced Gaussian kernel that is a finite-order Taylor expansion of a decomposed Gaussian kernel. The corresponding reduced Gaussian kernel least-mean-square (RGKLMS) algorithm is derived. The proposed algorithm avoids the sustained growth of the weighted network in a nonstationary environment via an implicit feature map. To verify the performance of the proposed algorithm, extensive simulations are conducted based on scenarios involving time-series prediction and nonlinear channel equalization, thereby proving that the RGKLMS algorithm is a universal approximator under suitable conditions. The simulation results also demonstrate that the RGKLMS algorithm can exhibit a comparable steady-state mean-square-error performance with a much lower computational complexity compared with other algorithms. [ABSTRACT FROM AUTHOR]

Support Vector Machine (SVM) is one of the successful methods of machine learning. SVM uses kernel tricks to efficiently learn non-linear classification tasks. The kernel function is applied to each data instance to map the original non-linear observations into a higher-dimensional space in which they become separable. The Gaussian kernel in Radial-Basis Function (RBF) is a popular kernel function used in various kernelized learning methods. In the RBF kernel, the sigma parameter (bandwidth of kernel function) should be fine-tuned, which is a challenging job. If the sigma value is very small, the decision boundary is highly non-linear. On the other hand, if the sigma value is large, the decision boundary tends to be linear. In this paper, an efficient kernel method called the Apollonius SVM Kernel (ASVMK) is introduced. Our algorithm adjusts the width of the kernel based on the data density and the representativeness of each class. The main idea is to introduce a kernel without adjusting the bandwidth of the kernel based on landmark points and the Apollonius similarity function. ASVMK finds the correlation of dense points and their neighbors in the data and tries to map the original non-linear observations in the highest dimensional space from these features. Reducing memory problems and low computations due to the shrinkage of the gram matrix are some of the benefits of the proposed kernel. The proposed method outperformed the state-of-the-art methods by up to 11%. • Find the width of the kernel neighborhood without the need to adjust the parameter. • The ASVM kernel is used in various datasets according to their density. • Hybridization of ASVMs with Density Peak Clustering to map the data to high space. • The kernel of ASVM is compatible with different datasets of various forms. [ABSTRACT FROM AUTHOR]

Feature selection can directly ascertain causes of faults by selecting useful features for fault diagnosis, which can simplify the procedures of fault diagnosis. As an efficient feature selection method, the linear kernel support vector machine recursive feature elimination (SVM-RFE) has been successfully applied to fault diagnosis. However, fault diagnosis is not a linear issue. Thus, this paper introduces the Gaussian kernel SVM-RFE to extract nonlinear features for fault diagnosis. The key issue is the selection of the kernel parameter for the Gaussian kernel SVM-RFE. We introduce three classical and simple kernel parameter selection methods and compare them in experiments. The proposed fault diagnosis framework combines the Gaussian kernel SVM-RFE and the SVM classifier, which can improve the performance of fault diagnosis. Experimental results on the Tennessee Eastman process indicate that the proposed framework for fault diagnosis is an advanced technique. [ABSTRACT FROM AUTHOR]

Attribute weight assignment plays a key role in multiple attribute decision making (MADM). For the issue of labeled multiple attribute decision making (LMADM), the existing methods of attribute weight determination that have been well developed for MADM usually ignore or do not take full advantage of the supervisory function of labels. As a result, the weights produced by these methods may not be ideal in practice. To make up for this deficiency, this paper develops an objective method based on Bayes risk. Specifically, the LMADM problem is first put forward, then a Gaussian kernel based loss function is proposed to cope with the drawback that the loss function in Bayes risk is usually determined by experts. Meanwhile, Mahalanobis distance and fuzzy neighborhood relationship are employed to measure the fuzziness of data set. Finally, a number of experiments, including the comparison experiments on UCI data and the effectiveness evaluation of fighter, are carried out to illustrate the superiority and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we present an algorithm for clustering based on univariate kernel density estimation, named ClusterKDE. It consists of an iterative procedure that in each step a new cluster is obtained by minimizing a smooth kernel function. Although in our applications we have used the univariate Gaussian kernel, any smooth kernel function can be used. The proposed algorithm has the advantage of not requiring a priori the number of cluster. Furthermore, the ClusterKDE algorithm is very simple, easy to implement, well-defined and stops in a finite number of steps, namely, it always converges independently of the initial point. We also illustrate our findings by numerical experiments which are obtained when our algorithm is implemented in the software Matlab and applied to practical applications. The results indicate that the ClusterKDE algorithm is competitive and fast when compared with the well-known Clusterdata and K-means algorithms, used by Matlab to clustering data. [ABSTRACT FROM PUBLISHER]

According to the industrialization of cities and the apparent increase in pollutants and greenhouse gases, the importance of forests as the natural lungs of the earth is felt more than ever to clean these pollutants. Annually, a large part of the forests is destroyed due to the lack of timely action during the fire. Knowledge about areas with a high-risk of fire and equipping these areas by constructing access routes and allocating the fire-fighting equipment can help to eliminate the destruction of the forest. In this research, the fire risk of region was forecasted and the risk map of that was provided using MODIS images by applying geographically weighted regression model with Gaussian kernel and ordinary least squares over the effective parameters in forest fire including distance from residential areas, distance from the river, distance from the road, height, slope, aspect, soil type, land use, average temperature, wind speed, and rainfall. After the evaluation, it was found that the geographically weighted regression model with Gaussian kernel forecasted 93.4% of the all fire points properly, however the ordinary least squares method could forecast properly only 66% of the fire points. [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]

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]

This paper presents a novel hierarchical clustering method using support vector machines. A common approach for hierarchical clustering is to use distance for the task. However, different choices for computing inter-cluster distances often lead to fairly distinct clustering outcomes, causing interpretation difficulties in practice. In this paper, we propose to use a one-class support vector machine (OC-SVM) to directly find high-density regions of data. Our algorithm generates nested set estimates using the OC-SVM and exploits the hierarchical structure of the estimated sets. We demonstrate the proposed algorithm on synthetic datasets. The cluster hierarchy is visualized with dendrograms and spanning trees [ABSTRACT FROM AUTHOR]

Abstract: As one of the methods to solve one-class classification problems (OCC), one-class support vector machines (OCSVM) have been applied to fault detection in recent years. Among all the kernels available for OCSVM, the Gaussian kernel is the most commonly used one. The selection of Gaussian kernel parameters influences greatly the performances of classifiers, which remains as an open problem. In this paper two methods are proposed to select Gaussian kernel parameters in OCSVM: according to the first one, the parameters are selected using the information of the farthest and the nearest neighbors of each sample; using the second one, the parameters are determined via detecting the “tightness” of the decision boundaries. The two proposed methods are tested on UCI data sets and Tennessee Eastman Process benchmark data sets. The results show that, the two proposed methods can be used to select suitable parameters for the Gaussian kernel, enabling the resulting OCSVM models to perform well on fault detection. [Copyright &y& Elsevier]

A direct implementation of the bilateral filter requires O(\sigmas^{2}) operations per pixel, where \sigmas is the (effective) width of the spatial kernel. A fast implementation of the bilateral filter that required O(1) operations per pixel with respect to \sigmas was recently proposed. This was done by using trigonometric functions for the range kernel of the bilateral filter, and by exploiting their so-called shiftability property. In particular, a fast implementation of the Gaussian bilateral filter was realized by approximating the Gaussian range kernel using raised cosines. Later, it was demonstrated that this idea could be extended to a larger class of filters, including the popular non-local means filter. As already observed, a flip side of this approach was that the run time depended on the width \sigmar of the range kernel. For an image with dynamic range [0,T], the run time scaled as O(T^2/\sigma^2r) with \sigmar. This made it difficult to implement narrow range kernels, particularly for images with large dynamic range. In this paper, we discuss this problem, and propose some simple steps to accelerate the implementation, in general, and for small \sigmar in particular. We provide some experimental results to demonstrate the acceleration that is achieved using these modifications. [ABSTRACT FROM PUBLISHER]

Abstract: The problem of multivariate alarm analysis and rationalization is complex and important in the area of smart alarm management due to the interrelationships between variables. The technique of capturing and visualizing the correlation information, especially from historical alarm data directly, is beneficial for further analysis. In this paper, the Gaussian kernel method is applied to generate pseudo continuous time series from the original binary alarm data. This can reduce the influence of missed, false, and chattering alarms. By taking into account time lags between alarm variables, a correlation color map of the transformed or pseudo data is used to show clusters of correlated variables with the alarm tags reordered to better group the correlated alarms. Thereafter correlation and redundancy information can be easily found and used to improve the alarm settings; and statistical methods such as singular value decomposition techniques can be applied within each cluster to help design multivariate alarm strategies. Industrial case studies are given to illustrate the practicality and efficacy of the proposed method. This improved method is shown to be better than the alarm similarity color map when applied in the analysis of industrial alarm data. [Copyright &y& Elsevier]

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]

We give several properties of the reproducing kernel Hilbert space induced by the Gaussian kernel, along with their implications for recent results in the complexity of the regularized least square algorithm in learning theory. [ABSTRACT FROM AUTHOR]

Abstract: This work is about intra-sentence segmentation performed before syntactic analysis of long sentences composed of at least 20 words in an English–Korean machine translation system. A long sentence has been known to spend enormous computational time and space when it is analyzed syntactically. It can also produce poor translation results. To resolve this problem, we partitioned a long sentence into a few segments to analyze each segment separately. To partition the sentence, firstly, we tried to find candidates for each segment position in the sentence. We then generated input vectors representing lexical contexts of the corresponding candidates and also used the support vector machines (SVM) algorithm to learn and recognize the appropriate segment positions. We used three kernel functions, the linear kernel, the polynomial kernel and the Gaussian kernel, to find optimal hyperplanes classifying proper positions and we compared results obtained from each kernel function. As a result of the experiments, we acquired 0.81, 0.83, and 0.79 f-measure values from the linear, polynomial and Gaussian kernel, respectively. [Copyright &y& Elsevier]

Gaussian kernels with flexible variances provide a rich family of Mercer kernels for learning algorithms. We show that the union of the unit balls of reproducing kernel Hilbert spaces generated by Gaussian kernels with flexible variances is a uniform Glivenko-Cantelli (uGC) class. This result confirms a conjecture concerning learnability of Gaussian kernels and verifies the uniform convergence of many learning algorithms involving Gaussians with changing variances. Rademacher averages and empirical covering numbers are used to estimate sample errors of multi-kernel regularization schemes associated with general loss functions. It is then shown that the regularization error associated with the least square loss and the Gaussian kernels can be greatly improved when flexible variances are allowed. Finally, for regularization schemes generated by Gaussian kernels with flexible variances we present explicit learning rates for regression with least square loss and classification with hinge loss. [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]

• Support Vector Data Description (SVDD) is a popular kernel-based unsupervised one-class classification method. The Gaussian kernel is the most common used kernel. • The Gaussian kernel has a tuning parameter, the kernel bandwidth, and it is important to choose it correctly. • We propose an automated, unsupervised, bandwidth selection method for SVDD. • Our proposed bandwidth is also appropriate for selecting the bandwidth for One Class Support Vector Machines (OCSVM). Support vector data description (SVDD) is a popular anomaly detection technique. The computation of the SVDD classifier requires a kernel function, for which the Gaussian kernel is a common choice. The Gaussian kernel has a bandwidth parameter, and it is important to set the value of this parameter correctly to ensure good results. A small bandwidth leads to overfitting, and the resulting SVDD classifier overestimates the number of anomalies, whereas a large bandwidth leads to underfitting and an inability to detect many anomalies. In this paper, we present a new, unsupervised method for selecting the Gaussian kernel bandwidth. Our method exploits a low-rank representation of the kernel matrix to suggest a kernel bandwidth value. Our new technique is competitive with the current state of the art for low-dimensional data and performs extremely well for many classes of high-dimensional data. This method is also applicable to one-class support vector machines (OCSVM). [ABSTRACT FROM AUTHOR]

To realize real time object tracking in complex environments, a kernel based MIL (KMIL) algorithm is proposed. The KMIL employs the Gaussian kernel function to deal with the inner product used in the weighted MIL (WMIL) algorithm. The method avoids computing the pos-likely-hood and neg-likely-hood many times, which results in a much faster tracker. To track an object with different motion, the searching areas for cropping the instances are varied according to the object’s size. Furthermore, an adaptive classifier updating strategy is presented to handle with the occlusion, pose variations and illumination changes. A similar score range is defined with respect to two given thresholds and a similar score from the second frame. Then, the learning rate will be set to be a small value when a similar score is out of the range. In contrast, a big learning rate is used. Finally, we compare its performance with that of the state-of-art algorithms on several classical videos. The experimental results show that the presented KMIL algorithm is faster and robust to the partial occlusion, pose variations and illumination changes. [ABSTRACT FROM AUTHOR]