Bridging deep and multiple kernel learning: A review

Kernel methods and deep learning are two of the most currently remarkable machine learning techniques that have achieved great success in many applications. Kernel methods are powerful tools to capture nonlinear patterns behind data. They implicitly learn high (even infinite) dimensional nonlinear features in the reproducing kernel Hilbert space (RKHS) while making the computation tractable by leveraging the kernel trick. It is commonly agreed that the success of kernel methods is very much dependent on the choice of kernel. Multiple kernel learning (MKL) is one possible scheme that performs kernel combination and selection for a variety of learning tasks, such as classification, clustering, and dimensionality reduction. Deep learning models project input data through several layers of nonlinearity and learn different levels of abstraction. The composition of multiple layers of nonlinear functions can approximate a rich set of naturally occurring input-output dependencies. To bridge kernel methods and deep learning, deep kernel learning has been proven to be an effective method to learn complex feature representations by combining the nonparametric flexibility of kernel methods with the structural properties of deep learning. This article presents a comprehensive overview of the state-of-the-art approaches that bridge the MKL and deep learning techniques. Specifically, we systematically review the typical hybrid models, training techniques, and their theoretical and practical benefits, followed by remaining challenges and future directions. We hope that our perspectives and discussions serve as valuable references for new practitioners and theoreticians seeking to innovate in the applications of the approaches incorporating the advantages of both paradigms and exploring new synergies.     

Parallel multiple kernel learning: a hybrid alternating direction method of multipliers

Multiple kernel learning (MKL) has recently become a hot topic in kernel methods. However, many MKL algorithms suffer from high computational cost. Moreover, standard MKL algorithms face the challenge of the rapid development of distributed computational environment such as cloud computing. In this study, a framework for parallel multiple kernel learning (PMKL) using hybrid alternating direction method of multipliers (H-ADMM) is developed to integrate the MKL algorithms and the multiprocessor system. The global problem with multiple kernel is divided into multiple local problems each of which is optimized in a local processor with a single kernel. An H-ADMM is proposed to make the local processors coordinate with each other to achieve the global optimal solution. The results of computational experiments show that PMKL exhibits high classification accuracy and fast computational speed.     

Non-uniform multiple kernel learning with cluster-based gating functions

Recently, multiple kernel learning (MKL) has gained increasing attention due to its empirical superiority over traditional single kernel based methods. However, most of state-of-the-art MKL methods are “uniform” in the sense that the relative weights of kernels keep fixed among all data.Here we propose a “non-uniform” MKL method with a data-dependent gating mechanism, i.e., adaptively determine the kernel weights for the samples. We utilize a soft clustering algorithm and then tune the weight for each cluster under the graph embedding (GE) framework. The idea of exploiting cluster structures is based on the observation that data from the same cluster tend to perform consistently, which thus increases the resistance to noises and results in more reliable estimate. Moreover, it is computationally simple to handle out-of-sample data, whose implicit RKHS representations are modulated by the posterior to each cluster.Quantitative studies between the proposed method and some representative MKL methods are conducted on both synthetic and widely used public data sets. The experimental results well validate its superiorities.     

Multiple kernel learning with gaussianity measures

Kernel methods are known to be effective for nonlinear multivariate analysis. One of the main issues in the practical use of kernel methods is the selection of kernel. There have been a lot of studies on kernel selection and kernel learning. Multiple kernel learning (MKL) is one of the promising kernel optimization approaches. Kernel methods are applied to various classifiers including Fisher discriminant analysis (FDA). FDA gives the Bayes optimal classification axis if the data distribution of each class in the feature space is a gaussian with a shared covariance structure. Based on this fact, an MKL framework based on the notion of gaussianity is proposed. As a concrete implementation, an empirical characteristic function is adopted to measure gaussianity in the feature space associated with a convex combination of kernel functions, and two MKL algorithms are derived. From experimental results on some data sets, we show that the proposed kernel learning followed by FDA offers strong classification power.     

半监督多示例核

在多示例学习中引入利用未标记示例的机制,能降低训练的成本并提高学习器的泛化能力。当前半监督多示例学习算法大部分是基于对包中的每一个示例进行标记,把多示例学习转化为一个单示例半监督学习问题。考虑到包的类标记由包中示例及包的结构决定,提出一种直接在包层次上进行半监督学习的多示例学习算法。通过定义多示例核,利用所有包(有标记和未标记)计算包层次的图拉普拉斯矩阵,作为优化目标中的光滑性惩罚项。在多示例核所张成的RKHS空间中寻找最优解被归结为确定一个经过未标记数据修改的多示例核函数,它能直接用在经典的核学习方法上。在实验数据集上对算法进行了测试,并和已有的算法进行了比较。实验结果表明,基于半监督多示例核的算法能够使用更少量的训练数据而达到与监督学习算法同样的精度,在有标记数据集相同的情况下利用未标记数据能有效地提高学习器的泛化能力。     

Multiple empirical kernel learning based on local information

The traditional multiple kernel learning (MKL) is usually based on implicit kernel mapping and adopts a certain combination of kernels instead of a single kernel. MKL has been demonstrated to have a significant advantage to the single-kernel learning. Although MKL sets different weights to different kernels, the weights are not changed over the whole input space. This weight setting might not been fit for those data with some underlying local distributions. In order to solve this problem, Gönen and Alpayd?n (2008) introduced a localizing gating model into the traditional MKL framework so as to assign different weights to a kernel in different regions of the input space. In this paper, we also integrate the localizing gating model into our previous work named MultiK-MHKS that is an effective multiple empirical kernel learning. In doing so, we can get multiple localized empirical kernel learning named MLEKL. Our contribution is that we first establish a localized formulation in the empirical kernel learning framework. The experimental results on benchmark data sets validate the effectiveness of the proposed MLEKL.     

通用稀疏多核学习

针对L1范数多核学习方法产生核权重的稀疏解时可能会导致有用信息的丢失和泛化性能退化,Lp范数多核学习方法产生核权重的非稀疏解时会产生很多冗余信息并对噪声敏感,提出了一种通用稀疏多核学习方法。该算法是基于L1范数和Lp范数(p>1) 混合的网状正则化多核学习方法,不仅能灵活的调整稀疏性,而且鼓励核权重的组效应,L1范数和Lp范数多核学习方法可以认为是该方法的特例。该方法引进的混合约束为非线性约束,故对此约束采用二阶泰勒展开式近似,并使用半无限规划来求解该优化问题。实验结果表明,改进后的方法在动态调整稀疏性的前提下能获得较好的分类性能,同时也支持组效应,从而验证了改进后的方法是有效可行的。     

模糊多核支持向量机研究进展

模糊多核支持向量机将模糊支持向量机与多核学习方法结合,通过构造隶属度函数和利用多个核函数的组合形式有效缓解了传统支持向量机模型对噪声数据敏感和多源异构数据学习困难等问题,广泛应用于模式识别和人工智能领域.综述了模糊多核支持向量机的理论基础及其研究现状,详细介绍模糊多核支持向量机中的关键问题,即模糊隶属度函数设计与多核学习方法,最后对模糊多核支持向量机算法未来的研究进行展望.     

An efficient multiple-kernel learning for pattern classification

Support vector machines (SVMs) have been broadly applied to classification problems. However, a successful application of SVMs depends heavily on the determination of the right type and suitable hyperparameter settings of kernel functions. Recently, multiple-kernel learning (MKL) algorithms have been developed to deal with these issues by combining different kernels together. The weight with each kernel in the combination is obtained through learning. Lanckriet et al. proposed a way of deriving the weights by transforming the learning into a semidefinite programming (SDP) problem with a transduction setting. However, the amount of time and space required by this method is demanding. In this paper, we reformulate the SDP problem with an induction setting and incorporate two strategies to reduce the search complexity of the learning process, based on the comments discussed in the Lanckriet et al. paper. The primal and dual forms of SDP are derived. A discussion on computation complexity is given. Experimental results obtained from synthetic and benchmark datasets show that the proposed method runs efficiently in multiple-kernel learning.     

Image category learning and classification via optimal linear combination of multiple partially matching kernels

Multiple kernel learning (MKL) aims at simultaneously optimizing kernel weights while training the support vector machine (SVM) to get satisfactory classification or regression results. Recent publications and developments based on SVM have shown that by using MKL one can enhance interpretability of the decision function and improve classifier performance, which motivates researchers to explore the use of homogeneous model obtained as linear combination of various types of kernels. In this paper, we show that MKL problems can be solved efficiently by modified projection gradient method and applied for image categorization and object detection. The kernel is defined as a linear combination of feature histogram function that can measure the degree of similarity of partial correspondence between feature sets for discriminative classification, which allows recognition robust to within-class variation, pose changes, and articulation. We evaluate our proposed framework on the ETH-80 dataset for several multi-level image encodings for supervised and unsupervised object recognition and report competitive results.     

Multiple kernel clustering based on centered kernel alignment

Multiple kernel clustering (MKC), which performs kernel-based data fusion for data clustering, is an emerging topic. It aims at solving clustering problems with multiple cues. Most MKC methods usually extend existing clustering methods with a multiple kernel learning (MKL) setting. In this paper, we propose a novel MKC method that is different from those popular approaches. Centered kernel alignment—an effective kernel evaluation measure—is employed in order to unify the two tasks of clustering and MKL into a single optimization framework. To solve the formulated optimization problem, an efficient two-step iterative algorithm is developed. Experiments on several UCI datasets and face image datasets validate the effectiveness and efficiency of our MKC algorithm.     

Integrated kernels and their properties

Kernel machines are widely considered to be powerful tools in various fields of information science. By using a kernel, an unknown target is represented by a function that belongs to a reproducing kernel Hilbert space (RKHS) corresponding to the kernel. The application area is widened by enlarging the RKHS such that it includes a wide class of functions. In this study, we demonstrate a method to perform this by using parameter integration of a parameterized kernel. Some numerical experiments show that the unresolved problem of finding a good parameter can be neglected.     

Design of semi-tensor product-based kernel function for SVM nonlinear classification

The kernel function method in support vector machine (SVM) is an excellent tool for nonlinear classification. How to design a kernel function is difficult for an SVM nonlinear classification problem, even for the polynomial kernel function. In this paper, we propose a new kind of polynomial kernel functions, called semi-tensor product kernel (STP-kernel), for an SVM nonlinear classification problem by semi-tensor product of matrix (STP) theory. We have shown the existence of the STP-kernel function and verified that it is just a polynomial kernel. In addition, we have shown the existence of the reproducing kernel Hilbert space (RKHS) associated with the STP-kernel function. Compared to the existing methods, it is much easier to construct the nonlinear feature mapping for an SVM nonlinear classification problem via an STP operator.     

A primal method for multiple kernel learning

The canonical support vector machines (SVMs) are based on a single kernel, recent publications have shown that using multiple kernels instead of a single one can enhance interpretability of the decision function and promote classification accuracy. However, most of existing approaches mainly reformulate the multiple kernel learning as a saddle point optimization problem which concentrates on solving the dual. In this paper, we show that the multiple kernel learning (MKL) problem can be reformulated as a BiConvex optimization and can also be solved in the primal. While the saddle point method still lacks convergence results, our proposed method exhibits strong optimization convergence properties. To solve the MKL problem, a two-stage algorithm that optimizes canonical SVMs and kernel weights alternately is proposed. Since standard Newton and gradient methods are too time-consuming, we employ the truncated-Newton method to optimize the canonical SVMs. The Hessian matrix need not be stored explicitly, and the Newton direction can be computed using several Preconditioned Conjugate Gradient steps on the Hessian operator equation, the algorithm is shown more efficient than the current primal approaches in this MKL setting. Furthermore, we use the Nesterov’s optimal gradient method to optimize the kernel weights. One remarkable advantage of solving in the primal is that it achieves much faster convergence rate than solving in the dual and does not require a two-stage algorithm even for the single kernel LapSVM. Introducing the Laplacian regularizer, we also extend our primal method to semi-supervised scenario. Extensive experiments on some UCI benchmarks have shown that the proposed algorithm converges rapidly and achieves competitive accuracy.     

一种改进的显性多核支持向量机

在支持向量机(Support vector machine, SVM)中, 对核函数的定义非常重要, 不同的核会产生不同的分类结果. 如何充分利用多个不同核函数的特点, 来共同提高SVM学习的效果, 已成为一个研究热点. 于是, 多核学习(Multiple kernel learning, MKL)方法应运而生. 最近, 有的学者提出了一种简单有效的稀疏MKL算法,即GMKL (Generalized MKL)算法, 它结合了L1 范式和L2范式的优点, 形成了一个对核权重的弹性限定. 然而, GMKL算法也并没有考虑到如何在充分利用已经选用的核函数中的共有信息. 另一方面, MultiK-MHKS算法则考虑了利用典型关联分析(Canonical correlation analysis, CCA)来获取核函数之间的共有信息, 但是却没有考虑到核函数的筛选问题. 本文模型则基于这两种算法进行了一定程度的改进, 我们称我们的算法为改进的显性多核支持向量机 (Improved domain multiple kernel support vector machine, IDMK-SVM). 我们证明了本文的模型保持了GMKL 的特性, 并且证明了算法的收敛性. 最后通过模拟实验, 本文证明了本文的多核学习方法相比于传统的多核学习方法有一定的精确性优势.     

一种基于SVM新的学习方法及应用

该文提出一种基于支持向量机的组合核函数的学习方法,它首先由遗传算法作为新的学习方法得到训练,组合核函数的权值在学习过程中被确定,并在决策模型的分类阶段用来作为参数。这种学习方法被应用在两个关于癌症诊断的公用数据集中,从而获得分类最优超平面。通过实验,这种学习方法显示出比用单一核函数具有较好的性能。     

一种基于样本加权的多尺度核支持向量机方法

多核学习方法是机器学习领域中的一个新的热点。核方法通过将数据映射到高维空间来增加线性分类器的计算能力,是目前解决非线性模式分析与分类问题的一种有效途径。但是在一些复杂的情况下,单个核函数构成的核学习方法并不能完全满足如数据异构或者不规则、样本规模大、样本分布不平坦等实际应用中的需求问题,因此将多个核函数进行组合以期获得更好的结果,是一种必然的发展趋势。因此提出一种基于样本加权的多尺度核支持向量机方法,通过不同尺度核函数对样本的拟合能力进行加权,从而得到基于样本加权的多尺度核支持向量机决策函数。通过在多个数据集上的实验分析可以得出所提方法对于各个数据集都获得了很高的分类准确率。     

基于中心核对齐的多核单类支持向量机

多核学习(MKL)方法在分类及回归任务中均取得了优于单核学习方法的性能,但传统的MKL方法均用于处理两类或多类分类问题.为了使MKL方法适用于处理单类分类(OCC)问题,提出了基于中心核对齐(CKA)的单类支持向量机(OCSVM).首先利用CKA计算每个核矩阵的权重,然后将所得权重用作线性组合系数,进而将不同类型的核函...     

Application of complex extreme learning machine to multiclass classification problems with high dimensionality: A THz spectra classification problem

We extend extreme learning machine (ELM) classifiers to complex Reproducing Kernel Hilbert Spaces (RKHS) where the input/output variables as well as the optimization variables are complex-valued. A new family of classifiers, called complex-valued ELM (CELM) suitable for complex-valued multiple-input–multiple-output processing is introduced. In the proposed method, the associated Lagrangian is computed using induced RKHS kernels, adopting a Wirtinger calculus approach formulated as a constrained optimization problem similarly to the conventional ELM classifier formulation. When training the CELM, the Karush–Khun–Tuker (KKT) theorem is used to solve the dual optimization problem that consists of satisfying simultaneously smallest training error as well as smallest norm of output weights criteria. The proposed formulation also addresses aspects of quaternary classification within a Clifford algebra context. For 2D complex-valued inputs, user-defined complex-coupled hyper-planes divide the classifier input space into four partitions. For 3D complex-valued inputs, the formulation generates three pairs of complex-coupled hyper-planes through orthogonal projections. The six hyper-planes then divide the 3D space into eight partitions. It is shown that the CELM problem formulation is equivalent to solving six real-valued ELM tasks, which are induced by projecting the chosen complex kernel across the different user-defined coordinate planes. A classification example of powdered samples on the basis of their terahertz spectral signatures is used to demonstrate the advantages of the CELM classifiers compared to their SVM counterparts. The proposed classifiers retain the advantages of their ELM counterparts, in that they can perform multiclass classification with lower computational complexity than SVM classifiers. Furthermore, because of their ability to perform classification tasks fast, the proposed formulations are of interest to real-time applications.     

Fast learning Circular Complex-valued Extreme Learning Machine (CC-ELM) for real-valued classification problems

In this paper, we present a fast learning fully complex-valued extreme learning machine classifier, referred to as ‘Circular Complex-valued Extreme Learning Machine (CC-ELM)’ for handling real-valued classification problems. CC-ELM is a single hidden layer network with non-linear input and hidden layers and a linear output layer. A circular transformation with a translational/rotational bias term that performs a one-to-one transformation of real-valued features to the complex plane is used as an activation function for the input neurons. The neurons in the hidden layer employ a fully complex-valued Gaussian-like (‘sech’) activation function. The input parameters of CC-ELM are chosen randomly and the output weights are computed analytically. This paper also presents an analytical proof to show that the decision boundaries of a single complex-valued neuron at the hidden and output layers of CC-ELM consist of two hyper-surfaces that intersect orthogonally. These orthogonal boundaries and the input circular transformation help CC-ELM to perform real-valued classification tasks efficiently.Performance of CC-ELM is evaluated using a set of benchmark real-valued classification problems from the University of California, Irvine machine learning repository. Finally, the performance of CC-ELM is compared with existing methods on two practical problems, viz., the acoustic emission signal classification problem and a mammogram classification problem. These study results show that CC-ELM performs better than other existing (both) real-valued and complex-valued classifiers, especially when the data sets are highly unbalanced.