Hidden space support vector machines

Hidden space support vector machines (HSSVMs) are presented in this paper. The input patterns are mapped into a high-dimensional hidden space by a set of hidden nonlinear functions and then the structural risk is introduced into the hidden space to construct HSSVMs. Moreover, the conditions for the nonlinear kernel function in HSSVMs are more relaxed, and even differentiability is not required. Compared with support vector machines (SVMs), HSSVMs can adopt more kinds of kernel functions because the positive definite property of the kernel function is not a necessary condition. The performance of HSSVMs for pattern recognition and regression estimation is also analyzed. Experiments on artificial and real-world domains confirm the feasibility and the validity of our algorithms.     

Support vector learning for fuzzy rule-based classification systems

To design a fuzzy rule-based classification system (fuzzy classifier) with good generalization ability in a high dimensional feature space has been an active research topic for a long time. As a powerful machine learning approach for pattern recognition problems, the support vector machine (SVM) is known to have good generalization ability. More importantly, an SVM can work very well on a high- (or even infinite) dimensional feature space. This paper investigates the connection between fuzzy classifiers and kernel machines, establishes a link between fuzzy rules and kernels, and proposes a learning algorithm for fuzzy classifiers. We first show that a fuzzy classifier implicitly defines a translation invariant kernel under the assumption that all membership functions associated with the same input variable are generated from location transformation of a reference function. Fuzzy inference on the IF-part of a fuzzy rule can be viewed as evaluating the kernel function. The kernel function is then proven to be a Mercer kernel if the reference functions meet a certain spectral requirement. The corresponding fuzzy classifier is named positive definite fuzzy classifier (PDFC). A PDFC can be built from the given training samples based on a support vector learning approach with the IF-part fuzzy rules given by the support vectors. Since the learning process minimizes an upper bound on the expected risk (expected prediction error) instead of the empirical risk (training error), the resulting PDFC usually has good generalization. Moreover, because of the sparsity properties of the SVMs, the number of fuzzy rules is irrelevant to the dimension of input space. In this sense, we avoid the "curse of dimensionality." Finally, PDFCs with different reference functions are constructed using the support vector learning approach. The performance of the PDFCs is illustrated by extensive experimental results. Comparisons with other methods are also provided.     

A kernel autoassociator approach to pattern classification.

Autoassociators are a special type of neural networks which, by learning to reproduce a given set of patterns, grasp the underlying concept that is useful for pattern classification. In this paper, we present a novel nonlinear model referred to as kernel autoassociators based on kernel methods. While conventional non-linear autoassociation models emphasize searching for the non-linear representations of input patterns, a kernel autoassociator takes a kernel feature space as the nonlinear manifold, and places emphasis on the reconstruction of input patterns from the kernel feature space. Two methods are proposed to address the reconstruction problem, using linear and multivariate polynomial functions, respectively. We apply the proposed model to novelty detection with or without novelty examples and study it on the promoter detection and sonar target recognition problems. We also apply the model to mclass classification problems including wine recognition, glass recognition, handwritten digit recognition, and face recognition. The experimental results show that, compared with conventional autoassociators and other recognition systems, kernel autoassociators can provide better or comparable performance for concept learning and recognition in various domains.     

Distance-Based Classifier via the Kernel Trick

For classifying large data sets, we propose a discriminant kernel that introduces a nonlinear mapping from the joint space of input data and output label to a discriminant space. Our method differs from traditional ones, which correspond to map nonlinearly from the input space to a feature space. The induced distance of our discriminant kernel is Eu- clidean and Fisher separable, as it is defined based on distance vectors of the feature space to distance vectors on the discriminant space. Unlike the support vector machines or the kernel Fisher discriminant analysis, the classifier does not need to solve a quadric program- ming problem or eigen-decomposition problems. Therefore, it is especially appropriate to the problems of processing large data sets. The classifier can be applied to face recognition, shape comparison and image classification benchmark data sets. The method is significantly faster than other methods and yet it can deliver comparable classification accuracy.     

黎曼流形上的保局投影在图像集匹配中的应用

目的提出了黎曼流形上局部结构特征保持的图像集匹配方法。方法该方法使用协方差矩阵建模图像集合,利用对称正定的非奇异协方差矩阵构成黎曼流形上的子空间,将图像集的匹配转化为流形上的点的匹配问题。通过基于协方差矩阵度量学习的核函数将黎曼流形上的协方差矩阵映射到欧几里德空间。不同于其他方法黎曼流形上的鉴别分析方法,考虑到样本分布的局部几何结构,引入了黎曼流形上局部保持的图像集鉴别分析方法,保持样本分布的局部邻域结构的同时提升样本的可分性。结果在基于图像集合的对象识别任务上测试了本文算法,在ETH80和YouTube Celebrities数据库分别进行了对象识别和人脸识别实验,分别达到91.5%和65.31%的识别率。结论实验结果表明,该方法取得了优于其他图像集匹配算法的效果。     

Latent Semantic Kernels

Kernel methods like support vector machines have successfully been used for text categorization. A standard choice of kernel function has been the inner product between the vector-space representation of two documents, in analogy with classical information retrieval (IR) approaches.Latent semantic indexing (LSI) has been successfully used for IR purposes as a technique for capturing semantic relations between terms and inserting them into the similarity measure between two documents. One of its main drawbacks, in IR, is its computational cost.In this paper we describe how the LSI approach can be implemented in a kernel-defined feature space.We provide experimental results demonstrating that the approach can significantly improve performance, and that it does not impair it.     

Preservation of local linearity by neighborhood subspace scaling for solving the pre-image problem

An important issue involved in kernel methods is the pre-image problem. However, it is an ill-posed problem, as the solution is usually nonexistent or not unique. In contrast to direct methods aimed at minimizing the distance in feature space, indirect methods aimed at constructing approximate equivalent models have shown outstanding performance. In this paper, an indirect method for solving the pre-image problem is proposed. In the proposed algorithm, an inverse mapping process is constructed based on a novel framework that preserves local linearity. In this framework, a local nonlinear transformation is implicitly conducted by neighborhood subspace scaling transformation to preserve the local linearity between feature space and input space. By extending the inverse mapping process to test samples, we can obtain pre-images in input space. The proposed method is non-iterative, and can be used for any kernel functions. Experimental results based on image denoising using kernel principal component analysis （PCA） show that the proposed method outperforms the state-of-the-art methods for solving the pre-image problem.     

Sparse least squares support vector training in the reduced empirical feature space

In this paper we discuss sparse least squares support vector machines (sparse LS SVMs) trained in the empirical feature space, which is spanned by the mapped training data. First, we show that the kernel associated with the empirical feature space gives the same value with that of the kernel associated with the feature space if one of the arguments of the kernels is mapped into the empirical feature space by the mapping function associated with the feature space. Using this fact, we show that training and testing of kernel-based methods can be done in the empirical feature space and that training of LS SVMs in the empirical feature space results in solving a set of linear equations. We then derive the sparse LS SVMs restricting the linearly independent training data in the empirical feature space by the Cholesky factorization. Support vectors correspond to the selected training data and they do not change even if the value of the margin parameter is changed. Thus for linear kernels, the number of support vectors is the number of input variables at most. By computer experiments we show that we can reduce the number of support vectors without deteriorating the generalization ability.

Local structure preservation in Kernel space for feature selection

For many machine learning and data mining tasks in the information explosion environment, one is often confronted with very high dimensional heterogeneous data. Demands for new methods to select discrimination and valuable features that are beneficial to classification and cluster have increased. In this paper, we propose a novel feature selection method to jointly map original data from input space to kernel space and conduct both subspace learning (via locality preserving projection) and feature selection (via a sparsity constraint). Specifically, the nonlinear relationship between data is explored adequately through mapping data from original low-dimensional space to kernel space. Meanwhile, the subspace learning technique is leveraged to preserve available information of local structure in ambient space. Last, by restricting the sparsity of the coefficient matrix, the weight of some features is 0. As a result, we eliminate redundant and irrelevant features and thus make our method select informative and distinguishing features. By comparing our proposed method with some state-of-the-art methods, the experimental results demonstrate that the proposed method outperformed the comparisons in terms of clustering task.

     

Fuzzy kernel perceptron

A new learning method, the fuzzy kernel perceptron (FKP), in which the fuzzy perceptron (FP) and the Mercer kernels are incorporated, is proposed in this paper. The proposed method first maps the input data into a high-dimensional feature space using some implicit mapping functions. Then, the FP is adopted to find a linear separating hyperplane in the high-dimensional feature space. Compared with the FP, the FKP is more suitable for solving the linearly nonseparable problems. In addition, it is also more efficient than the kernel perceptron (KP). Experimental results show that the FKP has better classification performance than FP, KP, and the support vector machine.     

核聚类算法

该文提出了一种用于聚类分析的核聚类方法，通过利用Mercer核，作者把输入空间的样本映射到高维特征空间后，在特征空间中进行聚类，由于经过了核函数的映射，使原来没有显现的特征突出来，从而能够更好地聚类，该核聚类方法在性能上比以典的聚类算法有较大的改进，具有更快的收敛速度以及更为准确的聚类，仿真实验的结果证实了核聚类方法的可行性和有效性。     

Spikernels: predicting arm movements by embedding population spike rate patterns in inner-product spaces

Inner-product operators, often referred to as kernels in statistical learning, define a mapping from some input space into a feature space. The focus of this letter is the construction of biologically motivated kernels for cortical activities. The kernels we derive, termed Spikernels, map spike count sequences into an abstract vector space in which we can perform various prediction tasks. We discuss in detail the derivation of Spikernels and describe an efficient algorithm for computing their value on any two sequences of neural population spike counts. We demonstrate the merits of our modeling approach by comparing the Spikernel to various standard kernels in the task of predicting hand movement velocities from cortical recordings. All of the kernels that we tested in our experiments outperform the standard scalar product used in linear regression, with the Spikernel consistently achieving the best performance.     

Feature space versus empirical kernel map and row kernel space in SVMs

In machine-learning technologies, the support vector machine (SV machine, SVM) is a brilliant invention with many merits, such as freedom from local minima, the widest possible margins separating different clusters, and a solid theoretical foundation. In this paper, we first explore the linear separability relationships between the high-dimensional feature space H and the empirical kernel map U as well as between H and the space of kernel outputs K. Second, we investigate the relations of the distances between separating hyperplanes and SVs in H and U, and derive an upper bound for the margin width in K. Third, as an application, we show experimentally that the separating hyperplane in H can be slightly adjusted through U. The experiments reveal that existing SVM training can linearly separate the data in H with considerable success. The results in this paper allow us to visualize the geometry of H by studying U and K.     

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.     

隐空间中参数化直接鉴别分析及其应用

提出了一种新的非线性特征抽取方法——隐空间中参数化直接鉴别分析。其主要思想是利用一核函数将原始输入空间非线性变换到隐空间,针对在该隐空间中类内散布矩阵总是奇异等问题,利用参数化直接鉴别分析进行特征抽取。与现有的核特征抽取方法不同的是,该方法不需要核函数满足Mercer 定理,从而增加了核函数的选择范围。更为重要的是,由于在隐空间中采用了参数化直接鉴别分析,不仅保留了参数化直接鉴别分析的优点,而且有效地抽取了样本的非线性特征;在该方法中提出了一个更为合理的加权系数矩阵,提高了分类性能。在FERET人脸数据库子库上的实验结果验证了该方法的有效性。     

Optimizing the kernel in the empirical feature space

In this paper, we present a method of kernel optimization by maximizing a measure of class separability in the empirical feature space, an Euclidean space in which the training data are embedded in such a way that the geometrical structure of the data in the feature space is preserved. Employing a data-dependent kernel, we derive an effective kernel optimization algorithm that maximizes the class separability of the data in the empirical feature space. It is shown that there exists a close relationship between the class separability measure introduced here and the alignment measure defined recently by Cristianini. Extensive simulations are carried out which show that the optimized kernel is more adaptive to the input data, and leads to a substantial, sometimes significant, improvement in the performance of various data classification algorithms.     

Estimating the support of a high-dimensional distribution.

Suppose you are given some data set drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S equals some a priori specified value between 0 and 1. We propose a method to approach this problem by trying to estimate a function f that is positive on S and negative on the complement. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length of the weight vector in an associated feature space. The expansion coefficients are found by solving a quadratic programming problem, which we do by carrying out sequential optimization over pairs of input patterns. We also provide a theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabeled data.     

On Learning and Cross-Validation with Decomposed Nyström Approximation of Kernel Matrix

The high computational costs of training kernel methods to solve nonlinear tasks limits their applicability. However, recently several fast training methods have been introduced for solving linear learning tasks. These can be used to solve nonlinear tasks by mapping the input data nonlinearly to a low-dimensional feature space. In this work, we consider the mapping induced by decomposing the Nyström approximation of the kernel matrix. We collect together prior results and derive new ones to show how to efficiently train, make predictions with and do cross-validation for reduced set approximations of learning algorithms, given an efficient linear solver. Specifically, we present an efficient method for removing basis vectors from the mapping, which we show to be important when performing cross-validation.     

Simple and robust methods for support vector expansions

Most support vector (SV) methods proposed in the recent literature can be viewed in a unified framework with great flexibility in terms of the choice of the kernel functions and their constraints. We show that all these problems can be solved within a unique approach if we are equipped with a robust method for finding a sparse solution of a linear system. Moreover, for such a purpose, we propose an iterative algorithm that can be simply implemented. Finally, we compare the classical SV approach with other, recently proposed, cross-correlation based, alternative methods. The simplicity of their implementation and the possibility of exactly calculating their computational complexity constitute important advantages in a real-time signal processing scenario