The University of Western Australia（西澳大利亚大学）；Karachi Institute of Economics and Technology（卡拉奇经济与技术学院）；Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）
Syed Muhammad Atif, Shujaat Khan, Imran Naseem, Roberto Togneri, Mohammed Bennamoun
A simple yet effective architectural design of radial basis function neural networks (RBFNN) makes them amongst the most popular conventional neural networks. The current generation of radial basis function neural network is equipped with multiple kernels which provide significant performance benefits compared to the previous generation using only a single kernel. In existing multi-kernel RBF algorithms, multi-kernel is formed by the convex combination of the base/primary kernels. In this paper, we propose a novel multi-kernel RBFNN in which every base kernel has its own (local) weight. This novel flexibility in the network provides better performance such as faster convergence rate, better local minima and resilience against stucking in poor local minima. These performance gains are achieved at a competitive computational complexity compared to the contemporary multi-kernel RBF algorithms. The proposed algorithm is thoroughly analysed for performance gain using mathematical and graphical illustrations and also evaluated on three different types of problems namely: (i) pattern classification, (ii) system identification and (iii) function approximation. Empirical results clearly show the superiority of the proposed algorithm compared to the existing state-of-the-art multi-kernel approaches.
Virginia Tech（弗吉尼亚理工大学）；Kuwait University（科威特大学）；Department of Industrial and Systems Engineering, Virginia Tech（弗吉尼亚理工大学工业与系统工程系）；Department of Industrial and Management Systems Engineering, Kuwait University（科威特大学工业与管理系统工程系）
Some response surface functions in complex engineering systems are usually highly nonlinear, unformed, and expensive-to-evaluate. To tackle this challenge, Bayesian optimization, which conducts sequential design via a posterior distribution over the objective function, is a critical method used to find the global optimum of black-box functions. Kernel functions play an important role in shaping the posterior distribution of the estimated function. The widely used kernel function, e.g., radial basis function (RBF), is very vulnerable and susceptible to outliers; the existence of outliers is cau…
Iqra University（伊克拉大学）；The University of Western Australia（西澳大学）
In this paper, we propose a novel adaptive kernel for the radial basis function (RBF) neural networks. The proposed kernel adaptively fuses the Euclidean and cosine distance measures to exploit the reciprocating properties of the two. The proposed framework dynamically adapts the weights of the participating kernels using the gradient descent method thereby alleviating the need for predetermined weights. The proposed method is shown to outperform the manual fusion of the kernels on three major problems of estimation namely nonlinear system identification, pattern classification and function ap…
University of Málaga（马拉加大学）；Biomedic Research Institute of Málaga (IBIMA)（马拉加生物医学研究所 (IBIMA)）；University of Michigan（密歇根大学）
Support Vector Machines (SVMs) are still one of the most popular and precise classifiers. The Radial Basis Function (RBF) kernel has been used in SVMs to separate among classes with considerable success. However, there is an intrinsic dependence on the initial value of the kernel hyperparameter. In this work, we propose OKSVM, an algorithm that automatically learns the RBF kernel hyperparameter and adjusts the SVM weights simultaneously. The proposed optimization technique is based on a gradient descent method. We analyze the performance of our approach with respect to the classical SVM for cl…
Computing Institute University of Campinas（坎皮纳斯大学计算研究所）；Facultad de Ciencias y Filosofía Universidad Peruana Cayetano Heredia（秘鲁卡耶塔诺埃雷迪亚大学科学与哲学学院）
SVM with an RBF kernel is usually one of the best classification algorithms for most data sets, but it is important to tune the two hyperparameters $C$ and $γ$ to the data itself. In general, the selection of the hyperparameters is a non-convex optimization problem and thus many algorithms have been proposed to solve it, among them: grid search, random search, Bayesian optimization, simulated annealing, particle swarm optimization, Nelder Mead, and others. There have also been proposals to decouple the selection of $γ$ and $C$. We empirically compare 18 of these proposed search algorithms (wit…
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We introduce and investigate matrix approximation by decomposition into a sum of radial basis function (RBF) components. An RBF component is a generalization of the outer product between a pair of vectors, where an RBF function replaces the scalar multiplication between individual vector elements. Even though the RBF functions are positive definite, the summation across components is not restricted to convex combinations and allows us to compute the decomposition for any real matrix that is not necessarily symmetric or positive definite. We formulate the problem of seeking such a decomposition…
The objective of this study is to investigate the efficient determination of $C$ and $γ$ for Support Vector Regression with RBF or mahalanobis kernel based on numerical and statistician considerations, which indicates the connection between $C$ and kernels and demonstrates that the deviation of geometric distance of neighbour observation in mapped space effects the predict accuracy of $ε$-SVR. We determinate the arrange of $γ$ & $C$ and propose our method to choose their best values.
Bayesian Optimization over Bounded Domains with the Beta Product Kernel
Research Unit of Health Sciences and Technology, Finland University of Oulu（芬兰健康科学与技术研究所，奥卢大学）；Department ESAT, Center for Processing Speech and Images KU Leuven（KU莱顿大学ESAT部门，图像与语音处理中心）
In the classical Gaussian SVM classification we use the feature space projection transforming points to normal distributions with fixed covariance matrices (identity in the standard RBF and the covariance of the whole dataset in Mahalanobis RBF). In this paper we add additional information to Gaussian SVM by considering local geometry-dependent feature space projection. We emphasize that our approach is in fact an algorithm for a construction of the new Gaussian-type kernel. We show that better (compared to standard RBF and Mahalanobis RBF) classification results are obtained in the simple cas…
We construct $\bf genRBF$ kernel, which generalizes the classical Gaussian RBF kernel to the case of incomplete data. We model the uncertainty contained in missing attributes making use of data distribution and associate every point with a conditional probability density function. This allows to embed incomplete data into the function space and to define a kernel between two missing data points based on scalar product in $L_2$. Experiments show that introduced kernel applied to SVM classifier gives better results than other state-of-the-art methods, especially in the case when large number of …
University of Electronic Science and Technology of China（电子科技大学）；Harbin Institute of Technology（哈尔滨工业大学）；Peng Cheng Lab（鹏城实验室）
Stein variational gradient descent (SVGD) and its variants have shown promising successes in approximate inference for complex distributions. In practice, we notice that the kernel used in SVGD-based methods has a decisive effect on the empirical performance. Radial basis function (RBF) kernel with median heuristics is a common choice in previous approaches, but unfortunately this has proven to be sub-optimal. Inspired by the paradigm of Multiple Kernel Learning (MKL), our solution to this flaw is using a combination of multiple kernels to approximate the optimal kernel, rather than a single o…
Kernel based methods have shown effective performance in many remote sensing classification tasks. However their performance significantly depend on its hyper-parameters. The conventional technique to estimate the parameter comes with high computational complexity. Thus, the objective of this letter is to propose an fast and efficient method to select the bandwidth parameter of the Gaussian kernel in the kernel based classification methods. The proposed method is developed based on the operators in the reproducing kernel Hilbert space and it is evaluated on Support vector machines and PerTurbo…
In deep learning models, learning more with less data is becoming more important. This paper explores how neural networks with normalized Radial Basis Function (RBF) kernels can be trained to achieve better sample efficiency. Moreover, we show how this kind of output layer can find embedding spaces where the classes are compact and well-separated. In order to achieve this, we propose a two-phase method to train those type of neural networks on classification tasks. Experiments on CIFAR-10 and CIFAR-100 show that networks with normalized kernels as output layer can achieve higher sample efficie…
In this paper, we compare 5 different nonlinear kernels: min-max, RBF, fRBF (folded RBF), acos, and acos-$χ^2$, on a wide range of publicly available datasets. The proposed fRBF kernel performs very similarly to the RBF kernel. Both RBF and fRBF kernels require an important tuning parameter ($γ$). Interestingly, for a significant portion of the datasets, the min-max kernel outperforms the best-tuned RBF/fRBF kernels. The acos kernel and acos-$χ^2$ kernel also perform well in general and in some datasets achieve the best accuracies. One crucial issue with the use of nonlinear kernels is the exc…
University of California, Davis（加州大学戴维斯分校）；Duke University（杜克大学）
We study the behavior of two kernel based sensor fusion algorithms, nonparametric canonical correlation analysis (NCCA) and alternating diffusion (AD), under the nonnull setting that the clean datasets collected from two sensors are modeled by a common low dimensional manifold embedded in a high dimensional Euclidean space and the datasets are corrupted by high dimensional noise. We establish the asymptotic limits and convergence rates for the eigenvalues of the associated kernel matrices assuming that the sample dimension and sample size are comparably large, where NCCA and AD are conducted u…
The deep neural network suffers from many fundamental issues in machine learning. For example, it often gets trapped into a local minimum in training, and its prediction uncertainty is hard to be assessed. To address these issues, we propose the so-called kernel-expanded stochastic neural network (K-StoNet) model, which incorporates support vector regression (SVR) as the first hidden layer and reformulates the neural network as a latent variable model. The former maps the input vector into an infinite dimensional feature space via a radial basis function (RBF) kernel, ensuring absence of local…
We present an approximation scheme for support vector machine models that use an RBF kernel. A second-order Maclaurin series approximation is used for exponentials of inner products between support vectors and test instances. The approximation is applicable to all kernel methods featuring sums of kernel evaluations and makes no assumptions regarding data normalization. The prediction speed of approximated models no longer relates to the amount of support vectors but is quadratic in terms of the number of input dimensions. If the number of input dimensions is small compared to the amount of sup…
Kernel ridge regression is well-known to achieve minimax optimal rates in low-dimensional settings. However, its behavior in high dimensions is much less understood. Recent work establishes consistency for kernel regression under certain assumptions on the ground truth function and the distribution of the input data. In this paper, we show that the rotational invariance property of commonly studied kernels (such as RBF, inner product kernels and fully-connected NTK of any depth) induces a bias towards low-degree polynomials in high dimensions. Our result implies a lower bound on the generaliza…
The GMM (generalized min-max) kernel was recently proposed (Li, 2016) as a measure of data similarity and was demonstrated effective in machine learning tasks. In order to use the GMM kernel for large-scale datasets, the prior work resorted to the (generalized) consistent weighted sampling (GCWS) to convert the GMM kernel to linear kernel. We call this approach as ``GMM-GCWS''. In the machine learning literature, there is a popular algorithm which we call ``RBF-RFF''. That is, one can use the ``random Fourier features'' (RFF) to convert the ``radial basis function'' (RBF) kernel to linear kern…
Shanghai Jiao Tong University（上海交通大学）；Institute of Image Processing and Pattern Recognition（图像处理与模式识别研究所）
This letter proposes a one-shot algorithm for feature-distributed kernel PCA. Our algorithm is inspired by the dual relationship between sample-distributed and feature-distributed scenario. This interesting relationship makes it possible to establish distributed kernel PCA for feature-distributed cases from ideas in distributed PCA in sample-distributed scenario. In theoretical part, we analyze the approximation error for both linear and RBF kernels. The result suggests that when eigenvalues decay fast, the proposed algorithm gives high quality results with low communication cost. This result …
The method of "random Fourier features (RFF)" has become a popular tool for approximating the "radial basis function (RBF)" kernel. The variance of RFF is actually large. Interestingly, the variance can be substantially reduced by a simple normalization step as we theoretically demonstrate. We name the improved scheme as the "normalized RFF (NRFF)". We also propose the "generalized min-max (GMM)" kernel as a measure of data similarity. GMM is positive definite as there is an associated hashing method named "generalized consistent weighted sampling (GCWS)" which linearizes this nonlinear kernel…
We propose and analyze a kernelized version of Q-learning. Although a kernel space is typically infinite-dimensional, extensive study has shown that generalization is only affected by the effective dimension of the data. We incorporate such ideas into the Q-learning framework and derive regret bounds for arbitrary kernels. In particular, we provide concrete bounds for linear kernels and Gaussian RBF kernels; notably, the latter bound looks almost identical to the former, only that the actual dimension is replaced by a different notion of dimensionality. Finally, we test our algorithm on a suit…
Conditional expectiles are becoming an increasingly important tool in finance as well as in other areas of applications. We analyse a support vector machine type approach for estimating conditional expectiles and establish learning rates that are minimax optimal modulo a logarithmic factor if Gaussian RBF kernels are used and the desired expectile is smooth in a Besov sense. As a special case, our learning rates improve the best known rates for kernel-based least squares regression in this scenario. Key ingredients of our statistical analysis are a general calibration inequality for the asymme…
Washington University in St. Louis（圣路易斯华盛顿大学）；Criteo（科韬）
Recent work in metric learning has significantly improved the state-of-the-art in k-nearest neighbor classification. Support vector machines (SVM), particularly with RBF kernels, are amongst the most popular classification algorithms that uses distance metrics to compare examples. This paper provides an empirical analysis of the efficacy of three of the most popular Mahalanobis metric learning algorithms as pre-processing for SVM training. We show that none of these algorithms generate metrics that lead to particularly satisfying improvements for SVM-RBF classification. As a remedy we introduc…
Aalto University（阿尔托大学）；Federal University of Ceará（塞阿拉联邦大学）
We show theoretical similarities between the Least Squares Support Vector Regression (LS-SVR) model with a Radial Basis Functions (RBF) kernel and maximum a posteriori (MAP) inference on Bayesian RBF networks with a specific Gaussian prior on the regression weights. Although previous works have pointed out similar expressions between those learning approaches, we explicit and formally state the existing correspondences. We empirically demonstrate our result by performing computational experiments with standard regression benchmarks. Our findings open a range of possibilities to improve LS-SVR …
Kernel functions in support vector machines (SVM) are needed to assess the similarity of input samples in order to classify these samples, for instance. Besides standard kernels such as Gaussian (i.e., radial basis function, RBF) or polynomial kernels, there are also specific kernels tailored to consider structure in the data for similarity assessment. In this article, we will capture structure in data by means of probabilistic mixture density models, for example Gaussian mixtures in the case of real-valued input spaces. From the distance measures that are inherently contained in these models,…
To enhance the accuracy of protein-protein interaction function prediction, a 2-order graphic neighbor information feature extraction method based on undirected simple graph is proposed in this paper, which extends the 1-order graphic neighbor featureextraction method. And the chi-square test statistical method is also involved in feature combination. To demonstrate the effectiveness of our 2-order graphic neighbor feature, four logistic regression models (logistic regression (abbrev. LR), diffusion kernel logistic regression (abbrev. DKLR), polynomial kernel logistic regression (abbrev. PKLR)…
The recently proposed "generalized min-max" (GMM) kernel can be efficiently linearized, with direct applications in large-scale statistical learning and fast near neighbor search. The linearized GMM kernel was extensively compared in with linearized radial basis function (RBF) kernel. On a large number of classification tasks, the tuning-free GMM kernel performs (surprisingly) well compared to the best-tuned RBF kernel. Nevertheless, one would naturally expect that the GMM kernel ought to be further improved if we introduce tuning parameters. In this paper, we study three simple constructions …
There are plenty of problems where the data available is scarce and expensive. We propose a generator of semi-artificial data with similar properties to the original data which enables development and testing of different data mining algorithms and optimization of their parameters. The generated data allow a large scale experimentation and simulations without danger of overfitting. The proposed generator is based on RBF networks, which learn sets of Gaussian kernels. These Gaussian kernels can be used in a generative mode to generate new data from the same distributions. To assess quality of t…
Sparse RBF Networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithms
Department of Mathematics, University of California, San Diego, CA 92093, United States（加州大学圣地亚哥分校数学系）；Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, United States（橡树岭国家实验室计算机科学与数学 division）
Standard kernels such as Matérn or RBF kernels only encode simple monotonic dependencies within the input space. Spectral mixture kernels have been proposed as general-purpose, flexible kernels for learning and discovering more complicated patterns in the data. Spectral mixture kernels have recently been generalized into non-stationary kernels by replacing the mixture weights, frequency means and variances by input-dependent functions. These functions have also been modelled as Gaussian processes on their own. In this paper we propose modelling the hyperparameter functions with neural networks…
The University of Sydney（悉尼大学）；University of Reading（雷丁大学）
A general framework of least squares support vector machine with low rank kernels, referred to as LR-LSSVM, is introduced in this paper. The special structure of low rank kernels with a controlled model size brings sparsity as well as computational efficiency to the proposed model. Meanwhile, a two-step optimization algorithm with three different criteria is proposed and various experiments are carried out using the example of the so-call robust RBF kernel to validate the model. The experiment results show that the performance of the proposed algorithm is comparable or superior to several exis…
Aalto University（阿尔托大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
Kernel functions are a powerful tool to enhance the $k$-means clustering algorithm via the kernel trick. It is known that the parameters of the chosen kernel function can have a dramatic impact on the result. In supervised settings, these can be tuned via cross-validation, but for clustering this is not straightforward and heuristics are usually employed. In this paper we study the impact of kernel parameters on kernel $k$-means. In particular, we derive a lower bound, tight up to constant factors, below which the parameter of the RBF kernel will render kernel $k$-means meaningless. We argue t…
Arizona State University（亚利桑那州立大学）；Lawrence Livermore National Labs（劳伦斯利弗莫尔国家实验室）；IBM T.J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Kernel fusion is a popular and effective approach for combining multiple features that characterize different aspects of data. Traditional approaches for Multiple Kernel Learning (MKL) attempt to learn the parameters for combining the kernels through sophisticated optimization procedures. In this paper, we propose an alternative approach that creates dense embeddings for data using the kernel similarities and adopts a deep neural network architecture for fusing the embeddings. In order to improve the effectiveness of this network, we introduce the kernel dropout regularization strategy coupled…
China University of Geosciences (Beijing)（中国地质大学（北京））；University of Naples Federico II（那不勒斯费德里科二世大学）
Heterogeneous data are commonly adopted as the inputs for some models that predict the future trends of some observations. Existing predictive models typically ignore the inconsistencies and imperfections in heterogeneous data while also failing to consider the (1) spatial correlations among monitoring points or (2) predictions for the entire study area. To address the above problems, this paper proposes a deep learning method for fusing heterogeneous data collected from multiple monitoring points using graph convolutional networks (GCNs) to predict the future trends of some observations and e…
School of Engineering, Dali University（大理大学工程学院）；RoyalFlush Information Network Co.,Ltd（同花顺信息网络有限公司）
A deep neural network for classification tasks is essentially consist of two components: feature extractors and function approximators. They usually work as an integrated whole, however, improvements on any components can promote the performance of the whole algorithm. This paper focus on designing a new function approximator. Conventionally, to build a function approximator, one usually uses the method based on the nonlinear activation function or the nonlinear kernel function and yields classical networks such as the feed-forward neural network (MLP) and the radial basis function network (RB…
We introduce a new family of non-linear neural network activation functions that mimic the properties induced by the widely-used Matérn family of kernels in Gaussian process (GP) models. This class spans a range of locally stationary models of various degrees of mean-square differentiability. We show an explicit link to the corresponding GP models in the case that the network consists of one infinitely wide hidden layer. In the limit of infinite smoothness the Matérn family results in the RBF kernel, and in this case we recover RBF activations. Matérn activation functions result in similar app…
Mammography is often used as the most common laboratory method for the detection of breast cancer, yet associated with the high cost and many side effects. Machine learning prediction as an alternative method has shown promising results. This paper presents a method based on a multilayer fuzzy expert system for the detection of breast cancer using an extreme learning machine (ELM) classification model integrated with radial basis function (RBF) kernel called ELM-RBF, considering the Wisconsin dataset. The performance of the proposed model is further compared with a linear-SVM model. The propos…
ZBW-Leibniz Information Centre for Economics（莱布尼茨经济信息中心）；Indian Institute of Technology Patna（印度理工学院巴特那分校）
Traditional Collaborative Filtering (CF) based methods are applied to understand the personal preferences of users/customers for items or products from the rating matrix. Usually, the rating matrix is sparse in nature. So there are some improved variants of the CF method that apply the increasing amount of side information to handle the sparsity problem. Only linear kernel or only non-linear kernel is applied in most of the available recommendation-related work to understand user-item latent feature embeddings from data. Only linear kernel or only non-linear kernel is not sufficient to learn c…
Gaussian processes are rich distributions over functions, with generalization properties determined by a kernel function. When used for long-range extrapolation, predictions are particularly sensitive to the choice of kernel parameters. It is therefore critical to account for kernel uncertainty in our predictive distributions. We propose a distribution over kernels formed by modelling a spectral mixture density with a Lévy process. The resulting distribution has support for all stationary covariances--including the popular RBF, periodic, and Matérn kernels--combined with inductive biases which…
Stanford University（斯坦福大学）；Duke University（杜克大学）；International Computer Science Institute（国际计算机科学研究所）；University of California, Berkeley（加州大学伯克利分校）
Kernel methods are widespread in machine learning; however, they are limited by the quadratic complexity of the construction, application, and storage of kernel matrices. Low-rank matrix approximation algorithms are widely used to address this problem and reduce the arithmetic and storage cost. However, we observed that for some datasets with wide intra-class variability, the optimal kernel parameter for smaller classes yields a matrix that is less well approximated by low-rank methods. In this paper, we propose an efficient structured low-rank approximation method -- the Block Basis Factoriza…
In this paper, we propose a special fusion method for combining ensembles of base classifiers utilizing new neural networks in order to improve overall efficiency of classification. While ensembles are designed such that each classifier is trained independently while the decision fusion is performed as a final procedure, in this method, we would be interested in making the fusion process more adaptive and efficient. This new combiner, called Neural Network Kernel Least Mean Square1, attempts to fuse outputs of the ensembles of classifiers. The proposed Neural Network has some special propertie…
