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In multiple instance learning problems, patterns are often given as bags and each bag consists of some instances. Most of existing research in the area focuses on multiple instance classification and multiple instance regression, while very limited work has been conducted for multiple instance clustering (MIC). This paper formulates a novel framework, maximum margin multiple instance clustering (M(3)IC), for MIC. However, it is impractical to directly solve the optimization problem of M(3)IC. Therefore, M(3)IC is relaxed in this paper to enable an efficient optimization solution with a combination of the constrained concave-convex procedure and the cutting plane method. Furthermore, this paper presents some important properties of the proposed method and discusses the relationship between the proposed method and some other related ones. An extensive set of empirical results are shown to demonstrate the advantages of the proposed method against existing research for both effectiveness and efficiency.

In safety critical applications, control architectures based on adaptive neural networks (NNs) must satisfy strict design specifications. This paper presents a practical approach for designing a mixed linear/adaptive model reference controller that recovers the performance of a reference model, and guarantees the boundedness of the tracking error within an a priori specified compact domain, in the presence of bounded uncertainties. The linear part of the controller results from the solution of an optimization problem where specifications are expressed as linear matrix inequality constraints. The linear controller is then augmented with a general adaptive NN that compensates for the uncertainties. The only requirement for the NN is that its output must be confined within pre-specified saturation limits. Toward this end a specific NN output confinement algorithm is proposed in this paper. The main advantages of the proposed approach are that requirements in terms of worst-case performance can be easily defined during the design phase, and that the design of the adaptation mechanism is largely independent from the synthesis of the linear controller. A numerical example is used to illustrate the design methodology.

Most partitioning algorithms iteratively partition a space into cells that contain underlying linear or nonlinear structures using linear partitioning strategies. The compactness of each cell depends on how well the (locally) linear partitioning strategy approximates the intrinsic structure. To partition a compact structure for complex data in a nonlinear context, this paper proposes a nonlinear partition strategy. This is a principal curve tree (PC-tree), which is implemented iteratively. Given that a PC passes through the middle of the data distribution, it allows for partitioning based on the arc length of the PC. To enhance the partitioning of a given space, a residual version of the PC-tree algorithm is developed, denoted here as the principal component analysis tree (PCR-tree) algorithm. Because of its residual property, the PCR-tree can yield the intrinsic dimension of high-dimensional data. Comparisons presented in this paper confirm that the proposed PC-tree and PCR-tree approaches show a better performance than several other competing partitioning algorithms in terms of vector quantization error and nearest neighbor search. The comparison also shows that the proposed algorithms outperform competing linear methods in total average coverage which measures the nonlinear compactness of partitioning algorithms.

This paper introduces the Clifford support vector machines (CSVM) as a generalization of the real and complex-valued support vector machines using the Clifford geometric algebra. In this framework, we handle the design of kernels involving the Clifford or geometric product. In this approach, one redefines the optimization variables as multivectors. This allows us to have a multivector as output. Therefore, we can represent multiple classes according to the dimension of the geometric algebra in which we work. We show that one can apply CSVM for classification and regression and also to build a recurrent CSVM. The CSVM is an attractive approach for the multiple input multiple output processing of high-dimensional geometric entities. We carried out comparisons between CSVM and the current approaches to solve multiclass classification and regression. We also study the performance of the recurrent CSVM with experiments involving time series. The authors believe that this paper can be of great use for researchers and practitioners interested in multiclass hypercomplex computing, particularly for applications in complex and quaternion signal and image processing, satellite control, neurocomputation, pattern recognition, computer vision, augmented virtual reality, robotics, and humanoids.

Support vector data description (SVDD) has become a very attractive kernel method due to its good results in many novelty detection problems. However, the decision function of SVDD is expressed in terms of the kernel expansion, which results in a run-time complexity linear in the number of support vectors. For applications where fast real-time response is needed, how to speed up the decision function is crucial. This paper aims at dealing with the issue of reducing the testing time complexity of SVDD. A method called fast SVDD (F-SVDD) is proposed. Unlike the traditional methods which all try to compress a kernel expansion into one with fewer terms, the proposed F-SVDD directly finds the preimage of a feature vector, and then uses a simple relationship between this feature vector and the SVDD sphere center to re-express the center with a single vector. The decision function of F-SVDD contains only one kernel term, and thus the decision boundary of F-SVDD is only spherical in the original space. Hence, the run-time complexity of the F-SVDD decision function is no longer linear in the support vectors, but is a constant, no matter how large the training set size is. In this paper, we also propose a novel direct preimage-finding method, which is noniterative and involves no free parameters. The unique preimage can be obtained in real time by the proposed direct method without taking trial-and-error. For demonstration, several real-world data sets and a large-scale data set, the extended MIT face data set, are used in experiments. In addition, a practical industry example regarding liquid crystal display micro-defect inspection is also used to compare the applicability of SVDD and our proposed F-SVDD when faced with mass data input. The results are very encouraging.

A crucial issue in designing learning machines is to select the correct model parameters. When the number of available samples is small, theoretical sample-based generalization bounds can prove effective, provided that they are tight and track the validation error correctly. The maximal discrepancy (MD) approach is a very promising technique for model selection for support vector machines (SVM), and estimates a classifier's generalization performance by multiple training cycles on random labeled data. This paper presents a general method to compute the generalization bounds for SVMs, which is based on referring the SVM parameters to an unsupervised solution, and shows that such an approach yields tight bounds and attains effective model selection. When one estimates the generalization error, one uses an unsupervised reference to constrain the complexity of the learning machine, thereby possibly decreasing sharply the number of admissible hypothesis. Although the methodology has a general value, the method described in the paper adopts vector quantization (VQ) as a representation paradigm, and introduces a biased regularization approach in bound computation and learning. Experimental results validate the proposed method on complex real-world data sets.

In this paper, a sparse learning algorithm, probabilistic classification vector machines (PCVMs), is proposed. We analyze relevance vector machines (RVMs) for classification problems and observe that adopting the same prior for different classes may lead to unstable solutions. In order to tackle this problem, a signed and truncated Gaussian prior is adopted over every weight in PCVMs, where the sign of prior is determined by the class label, i.e., +1 or -1. The truncated Gaussian prior not only restricts the sign of weights but also leads to a sparse estimation of weight vectors, and thus controls the complexity of the model. In PCVMs, the kernel parameters can be optimized simultaneously within the training algorithm. The performance of PCVMs is extensively evaluated on four synthetic data sets and 13 benchmark data sets using three performance metrics, error rate (ERR), area under the curve of receiver operating characteristic (AUC), and root mean squared error (RMSE). We compare PCVMs with soft-margin support vector machines (SVM(Soft)), hard-margin support vector machines (SVM(Hard)), SVM with the kernel parameters optimized by PCVMs (SVM(PCVM)), relevance vector machines (RVMs), and some other baseline classifiers. Through five replications of twofold cross-validation F test, i.e., 5 x 2 cross-validation F test, over single data sets and Friedman test with the corresponding post-hoc test to compare these algorithms over multiple data sets, we notice that PCVMs outperform other algorithms, including SVM(Soft), SVM(Hard), RVM, and SVM(PCVM), on most of the data sets under the three metrics, especially under AUC. Our results also reveal that the performance of SVM(PCVM) is slightly better than SVM(Soft), implying that the parameter optimization algorithm in PCVMs is better than cross validation in terms of performance and computational complexity. In this paper, we also discuss the superiority of PCVMs' formulation using maximum a posteriori (MAP) analysis and margin analysis, which explain the empirical success of PCVMs.

This paper presents a subgradient-based neural network to solve a nonsmooth nonconvex optimization problem with a nonsmooth nonconvex objective function, a class of affine equality constraints, and a class of nonsmooth convex inequality constraints. The proposed neural network is modeled with a differential inclusion. Under a suitable assumption on the constraint set and a proper assumption on the objective function, it is proved that for a sufficiently large penalty parameter, there exists a unique global solution to the neural network and the trajectory of the network can reach the feasible region in finite time and stay there thereafter. It is proved that the trajectory of the neural network converges to the set which consists of the equilibrium points of the neural network, and coincides with the set which consists of the critical points of the objective function in the feasible region. A condition is given to ensure the convergence to the equilibrium point set in finite time. Moreover, under suitable assumptions, the coincidence between the solution to the differential inclusion and the ldquoslow solutionrdquo of it is also proved. Furthermore, three typical examples are given to present the effectiveness of the theoretic results obtained in this paper and the good performance of the proposed neural network.

The self-organizing network (SON)-based multiple model system is a recently proposed method for identifying the dynamics of a general nonlinear system. It has been observed by researchers that cooperative learning among neighboring regions is sometimes important for the success of identification of a nonlinear system under the multiple model system framework. In this paper, we intend to formally evaluate the effects of cooperative learning and topology preservation in identification of multiple model system based on SON. The results of the mathematical analysis supports the heuristic that a good learning strategy for identifying the local model parameters of a SON-based multiple model system is to choose a neighborhood function whose effective region is initially wider and is reduced gradually during learning. An example of nonlinear function approximation is also provided at the end of this paper to demonstrate the results of the mathematical analysis.

As is well known in statistics, the resulting linear regressors by using the rank-based Wilcoxon approach to linear regression problems are usually robust against (or insensitive to) outliers. This motivates us to introduce in this paper the Wilcoxon approach to the area of machine learning. Specifically, we investigate four new learning machines, namely Wilcoxon neural network (WNN), Wilcoxon generalized radial basis function network (WGRBFN), Wilcoxon fuzzy neural network (WFNN), and kernel-based Wilcoxon regressor (KWR). These provide alternative learning machines when faced with general nonlinear learning problems. Simple weights updating rules based on gradient descent will be derived. Some numerical examples will be provided to compare the robustness against outliers for various learning machines. Simulation results show that the Wilcoxon learning machines proposed in this paper have good robustness against outliers. We firmly believe that the Wilcoxon approach will provide a promising methodology for many machine learning problems.

This paper is concerned with an enhanced independent component analysis (ICA) and its application to face recognition. Typically, face representations obtained by ICA involve unsupervised learning and high-order statistics. In this paper, we develop an enhancement of the generic ICA by augmenting this method by the Fisher linear discriminant analysis (LDA); hence, its abbreviation, FICA. The FICA is systematically developed and presented along with its underlying architecture. A comparative analysis explores four distance metrics, as well as classification with support vector machines (SVMs). We demonstrate that the FICA approach leads to the formation of well-separated classes in low-dimension subspace and is endowed with a great deal of insensitivity to large variation in illumination and facial expression. The comprehensive experiments are completed for the facial-recognition technology (FERET) face database; a comparative analysis demonstrates that FICA comes with improved classification rates when compared with some other conventional approaches such as eigenface, fisherface, and the ICA itself.

In this paper, we propose a new neural architecture for classification of visual patterns that is motivated by the two concepts of image pyramids and local receptive fields. The new architecture, called pyramidal neural network (PyraNet), has a hierarchical structure with two types of processing layers: Pyramidal layers and one-dimensional (1-D) layers. In the new network, nonlinear two-dimensional (2-D) neurons are trained to perform both image feature extraction and dimensionality reduction. We present and analyze five training methods for PyraNet [gradient descent (GD), gradient descent with momentum, resilient back-propagation (RPROP), Polak-Ribiere conjugate gradient (CG), and Levenberg-Marquadrt (LM)] and two choices of error functions [mean-square-error (mse) and cross-entropy (CE)]. In this paper, we apply PyraNet to determine gender from a facial image, and compare its performance on the standard facial recognition technology (FERET) database with three classifiers: The convolutional neural network (NN), the k-nearest neighbor (k-NN), and the support vector machine (SVM).

In this paper, an adaptive wavelet-network-based control approach is proposed for highly nonlinear uncertain dynamical systems. Wavelet network, as a kind of universal approximator, has two novel properties--orthonormality and multiresolution. The orthonormal property ensures that adding a new resolution (new wavelets) does not affect the existing wavelet network that may have been well tuned. In the sequel, the online adjustment of the structure of the nonlinear adaptive wavelet controller (AWC) can be done in a constructive manner by gradually increasing the network resolution. The multiresolution property, on the other hand, assures a guaranteed improvement of the approximation precision when a new resolution is added. In real life problems we are unable to know the adequate size of a network, either a neural network (NN) or a wavelet network, to produce the required approximation precision. By virtue of the novel wavelet network properties, a coarse or very simple structure can be selected first. If the system fails to converge after the elapse of a dwell time, a new wavelet resolution is considered to be necessary and added directly. In this manner, the AWC can be easily constructed and tuned from the coarse to finer levels until the performance requirement is satisfied. The trial and error way of selecting the network structure, which may lead to either an inadequate or a highly redundant structure, can be avoided. In this paper, the proposed adaptive wavelet network is applied first to a class of nonlinear dynamical systems with a partially known model and an affine-in-input structure. Then, the adaptive wavelet network is applied to a class of nonlinear nonaffine dynamical systems.

In dealing with large data sets, the reduced support vector machine (RSVM) was proposed for the practical objective to overcome some computational difficulties as well as to reduce the model complexity. In this paper, we study the RSVM from the viewpoint of sampling design, its robustness, and the spectral analysis of the reduced kernel. We consider the nonlinear separating surface as a mixture of kernels. Instead of a full model, the RSVM uses a reduced mixture with kernels sampled from certain candidate set. Our main results center on two major themes. One is the robustness of the random subset mixture model. The other is the spectral analysis of the reduced kernel. The robustness is judged by a few criteria as follows: 1) model variation measure; 2) model bias (deviation) between the reduced model and the full model; and 3) test power in distinguishing the reduced model from the full one. For the spectral analysis, we compare the eigenstructures of the full kernel matrix and the approximation kernel matrix. The approximation kernels are generated by uniform random subsets. The small discrepancies between them indicate that the approximation kernels can retain most of the relevant information for learning tasks in the full kernel. We focus on some statistical theory of the reduced set method mainly in the context of the RSVM. The use of a uniform random subset is not limited to the RSVM. This approach can act as a supplemental algorithm on top of a basic optimization algorithm, wherein the actual optimization takes place on the subset-approximated data. The statistical properties discussed in this paper are still valid.

In some pattern recognition tasks, the dimension of the sample space is larger than the number of samples in the training set. This is known as the "small sample size problem". Linear discriminant analysis (LDA) techniques cannot be applied directly to the small sample size case. The small sample size problem is also encountered when kernel approaches are used for recognition. In this paper, we attempt to answer the question of "How should one choose the optimal projection vectors for feature extraction in the small sample size case?" Based on our findings, we propose a new method called the kernel discriminative common vector method. In this method, we first nonlinearly map the original input space to an implicit higher dimensional feature space, in which the data are hoped to be linearly separable. Then, the optimal projection vectors are computed in this transformed space. The proposed method yields an optimal solution for maximizing a modified Fisher's linear discriminant criterion, discussed in the paper. Thus, under certain conditions, a 100% recognition rate is guaranteed for the training set samples. Experiments on test data also show that, in many situations, the generalization performance of the proposed method compares favorably with other kernel approaches.

A new adaptive backpropagation (BP) algorithm based on Lyapunov stability theory for neural networks is developed in this paper. It is shown that the candidate of a Lyapunov function V(k) of the tracking error between the output of a neural network and the desired reference signal is chosen first, and the weights of the neural network are then updated, from the output layer to the input layer, in the sense that deltaV(k) = V(k) - V(k - 1)0. The output tracking error can then asymptotically converge to zero according to Lyapunov stability theory. Unlike gradient-based BP training algorithms, the new Lyapunov adaptive BP algorithm in this paper is not used for searching the global minimum point along the cost-function surface in the weight space, but it is aimed at constructing an energy surface with a single global minimum point through the adaptive adjustment of the weights as the time goes to infinity. Although a neural network may have bounded input disturbances, the effects of the disturbances can be eliminated, and asymptotic error convergence can be obtained. The new Lyapunov adaptive BP algorithm is then applied to the design of an adaptive filter in the simulation example to show the fast error convergence and strong robustness with respect to large bounded input disturbances.

This paper investigates new learning algorithms (LF I and LF II) based on Lyapunov function for the training of feedforward neural networks. It is observed that such algorithms have interesting parallel with the popular backpropagation (BP) algorithm where the fixed learning rate is replaced by an adaptive learning rate computed using convergence theorem based on Lyapunov stability theory. LF II, a modified version of LF I, has been introduced with an aim to avoid local minima. This modification also helps in improving the convergence speed in some cases. Conditions for achieving global minimum for these kind of algorithms have been studied in detail. The performances of the proposed algorithms are compared with BP algorithm and extended Kalman filtering (EKF) on three bench-mark function approximation problems: XOR, 3-bit parity, and 8-3 encoder. The comparisons are made in terms of number of learning iterations and computational time required for convergence. It is found that the proposed algorithms (LF I and II) are much faster in convergence than other two algorithms to attain same accuracy. Finally, the comparison is made on a complex two-dimensional (2-D) Gabor function and effect of adaptive learning rate for faster convergence is verified. In a nutshell, the investigations made in this paper help us better understand the learning procedure of feedforward neural networks in terms of adaptive learning rate, convergence speed, and local minima.

In pattern recognition, feature extraction techniques are widely employed to reduce the dimensionality of data and to enhance the discriminatory information. Principal component analysis (PCA) and linear discriminant analysis (LDA) are the two most popular linear dimensionality reduction methods. However, PCA is not very effective for the extraction of the most discriminant features, and LDA is not stable due to the small sample size problem . In this paper, we propose some new (linear and nonlinear) feature extractors based on maximum margin criterion (MMC). Geometrically, feature extractors based on MMC maximize the (average) margin between classes after dimensionality reduction. It is shown that MMC can represent class separability better than PCA. As a connection to LDA, we may also derive LDA from MMC by incorporating some constraints. By using some other constraints, we establish a new linear feature extractor that does not suffer from the small sample size problem, which is known to cause serious stability problems for LDA. The kernelized (nonlinear) counterpart of this linear feature extractor is also established in the paper. Our extensive experiments demonstrate that the new feature extractors are effective, stable, and efficient.

The problem of optimal asymmetric Hopfield-type associative memory (HAM) design based on perceptron-type learning algorithms is considered. It is found that most of the existing methods considered the design problem as either 1) finding optimal hyperplanes according to normal distance from the prototype vectors to the hyperplane surface or 2) obtaining weight matrix W=[w/sub ij/] by solving a constraint optimization problem. In this paper, we show that since the state space of the HAM consists of only bipolar patterns, i.e., V=(v/sub 1/,v/sub 2/,...,v/sub N/)/sup T//spl isin/{-1,+1}/sup N/, the basins of attraction around each prototype (training) vector should be expanded by using Hamming distance measure. For this reason, in this paper, the design problem is considered from a different point of view. Our idea is to systematically increase the size of the training set according to the desired basin of attraction around each prototype vector. We name this concept the higher order Hamming stability and show that conventional minimum-overlap algorithm can be modified to incorporate this concept. Experimental results show that the recall capability as well as the number of spurious memories are all improved by using the proposed method. Moreover, it is well known that setting all self-connections w/sub ii//spl forall/i to zero has the effect of reducing the number of spurious memories in state space. From the experimental results, we find that the basin width around each prototype vector can be enlarged by allowing nonzero diagonal elements on learning of the weight matrix W. If the magnitude of w/sub ii/ is small for all i, then the condition w/sub ii/=0/spl forall/i can be relaxed without seriously affecting the number of spurious memories in the state space. Therefore, the method proposed in this paper can be used to increase the basin width around each prototype vector with the cost of slightly increasing the number of spurious memories in the state space.

A novel feature selection method using the concept of mutual information (MI) is proposed in this paper. In all MI based feature selection methods, effective and efficient estimation of high-dimensional MI is crucial. In this paper, a pruned Parzen window estimator and the quadratic mutual information (QMI) are combined to address this problem. The results show that the proposed approach can estimate the MI in an effective and efficient way. With this contribution, a novel feature selection method is developed to identify the salient features one by one. Also, the appropriate feature subsets for classification can be reliably estimated. The proposed methodology is thoroughly tested in four different classification applications in which the number of features ranged from less than 10 to over 15,000. The presented results are very promising and corroborate the contribution of the proposed feature selection methodology.

The decision-making process of many binary classification systems is based on the likelihood ratio (LR) scores of test patterns. This paper shows that LR scores can be expressed in terms of the similarity between the supervectors (SVs) formed by stacking the mean vectors of Gaussian mixture models corresponding to the test patterns, the target model, and the background model. By interpreting the support vector machine (SVM) kernels as a specific similarity (or discriminant) function between SVs, this paper shows that LR scoring is a special case of SVM scoring and that most sequence kernels can be obtained by assuming a specific form for the similarity function of SVs. This paper further shows that this assumption can be relaxed to derive a new general kernel. The kernel function is general in that it is a linear combination of any kernels belonging to the reproducing kernel Hilbert space. The combination weights are obtained by optimizing the ability of a discriminant function to separate the positive and negative classes using either regression analysis or SVM training. The idea was applied to both high-and low-level speaker verification. In both cases, results show that the proposed kernels achieve better performance than several state-of-the-art sequence kernels. Further performance enhancement was also observed when the high-level scores were combined with acoustic scores.

Recurrent neural networks for solving constrained least absolute deviation (LAD) problems or L(1)-norm optimization problems have attracted much interest in recent years. But so far most neural networks can only deal with some special linear constraints efficiently. In this paper, two neural networks are proposed for solving LAD problems with various linear constraints including equality, two-sided inequality and bound constraints. When tailored to solve some special cases of LAD problems in which not all types of constraints are present, the two networks can yield simpler architectures than most existing ones in the literature. In particular, for solving problems with both equality and one-sided inequality constraints, another network is invented. All of the networks proposed in this paper are rigorously shown to be capable of solving the corresponding problems. The different networks designed for solving the same types of problems possess the same structural complexity, which is due to the fact these architectures share the same computing blocks and only differ in connections between some blocks. By this means, some flexibility for circuits realization is provided. Numerical simulations are carried out to illustrate the theoretical results and compare the convergence rates of the networks.

This paper proposes an uncorrelated multilinear principal component analysis (UMPCA) algorithm for unsupervised subspace learning of tensorial data. It should be viewed as a multilinear extension of the classical principal component analysis (PCA) framework. Through successive variance maximization, UMPCA seeks a tensor-to-vector projection (TVP) that captures most of the variation in the original tensorial input while producing uncorrelated features. The solution consists of sequential iterative steps based on the alternating projection method. In addition to deriving the UMPCA framework, this work offers a way to systematically determine the maximum number of uncorrelated multilinear features that can be extracted by the method. UMPCA is compared against the baseline PCA solution and its five state-of-the-art multilinear extensions, namely two-dimensional PCA (2DPCA), concurrent subspaces analysis (CSA), tensor rank-one decomposition (TROD), generalized PCA (GPCA), and multilinear PCA (MPCA), on the tasks of unsupervised face and gait recognition. Experimental results included in this paper suggest that UMPCA is particularly effective in determining the low-dimensional projection space needed in such recognition tasks.

The concepts of permitted and forbidden sets enable a new perspective of the memory in neural networks. Such concepts exhibit interesting dynamics in recurrent neural networks. This paper studies the basic theories of permitted and forbidden sets of the linear threshold discrete-time recurrent neural networks. The linear threshold transfer function has been regarded as an adequate transfer function for recurrent neural networks. Networks with this transfer function form a class of hybrid analog and digital networks which are especially useful for perceptual computations. Networks in discrete time can directly provide algorithms for efficient implementation in digital hardware. The main contribution of this paper is to establish foundations of permitted and forbidden sets. Necessary and sufficient conditions for the linear threshold discrete-time recurrent neural networks are obtained for complete convergence, existence of permitted and forbidden sets, as well as conditionally multiattractivity, respectively. Simulation studies explore some possible interesting practical applications.

An unsupervised competitive neural network for efficient clustering of Gaussian probability density function (GPDF) data of continuous density hidden Markov models (CDHMMs) is proposed in this paper. The proposed unsupervised competitive neural network, called the divergence-based centroid neural network (DCNN), employs the divergence measure as its distance measure and utilizes the statistical characteristics of observation densities in the HMM for speech recognition problems. While the conventional clustering algorithms used for the vector quantization (VQ) codebook design utilize only the mean values of the observation densities in the HMM, the proposed DCNN utilizes both the mean and the covariance values. When compared with other conventional unsupervised neural networks, the DCNN successfully allocates more code vectors to the regions where GPDF data are densely distributed while it allocates fewer code vectors to the regions where GPDF data are sparsely distributed. When applied to Korean monophone recognition problems as a tool to reduce the size of the codebook, the DCNN reduced the number of GPDFs used for code vectors by 65.3% while preserving recognition accuracy. Experimental results with a divergence-based k-means algorithm and a divergence-based self-organizing map algorithm are also presented in this paper for a performance comparison.

This paper deals with the problem of multiuser detection in direct-sequence code-division multiple-access (DS-CDMA) systems in multipath environments. The existing multiuser detectors can be divided into two categories: (1) low-complexity poor-performance linear detectors and (2) high-complexity good-performance nonlinear detectors. In particular, in channels where the orthogonality of the code sequences is destroyed by multipath, detectors with linear complexity perform much worse than the nonlinear detectors. In this paper, we propose an enhanced multisurface method (EMSM) for multiuser detection in multipath channels. EMSM is an intermediate piecewise linear detection scheme with a run-time complexity linear in the number of users. Its bit error rate performance is compared with existing linear detectors, a nonlinear radial basis function detector trained by the new support vector learning algorithm, and Verdu's optimal detector. Simulations in multipath channels, for both synchronous and asynchronous cases, indicate that it always outperforms all other linear detectors, performing nearly as well as nonlinear detectors.

A novel radial basis function neural network for discriminant analysis is presented in this paper. In contrast to many other researches, this work focuses on the exploitation of the weight structure of radial basis function neural networks using the Bayesian method. It is expected that the performance of a radial basis function neural network with a well-explored weight structure can be improved. As the weight structure of a radial basis function neural network is commonly unknown, the Bayesian method is, therefore, used in this paper to study this a priori structure. Two weight structures are investigated in this study, i.e., a single-Gaussian structure and a two-Gaussian structure. An expectation-maximization learning algorithm is used to estimate the weights. The simulation results showed that the proposed radial basis function neural network with a weight structure of two Gaussians outperformed the other algorithms.

In this paper, we analyze Boolean functions using a recently proposed measure of their complexity. This complexity measure, motivated by the aim of relating the complexity of the functions with the generalization ability that can be obtained when the functions are implemented in feed-forward neural networks, is the sum of a number of components. We concentrate on the case in which we use the first two of these components. The first is related to the "average sensitivity" of the function and the second is, in a sense, a measure of the "randomness" or lack of structure of the function. In this paper, we investigate the importance of using the second term in the complexity measure, and we consider to what extent these two terms suffice as an indicator of how difficult it is to learn a Boolean function. We also explore the existence of very complex Boolean functions, considering, in particular, the symmetric Boolean functions.

The convergence of Oja's principal component analysis (PCA) learning algorithms is a difficult topic for direct study and analysis. Traditionally, the convergence of these algorithms is indirectly analyzed via certain deterministic continuous time (DCT) systems. Such a method will require the learning rate to converge to zero, which is not a reasonable requirement to impose in many practical applications. Recently, deterministic discrete time (DDT) systems have been proposed instead to indirectly interpret the dynamics of the learning algorithms. Unlike DCT systems, DDT systems allow learning rates to be constant (which can be a nonzero). This paper will provide some important results relating to the convergence of a DDT system of Oja's PCA learning algorithm. It has the following contributions: 1) A number of invariant sets are obtained, based on which we can show that any trajectory starting from a point in the invariant set will remain in the set forever. Thus, the nondivergence of the trajectories is guaranteed. 2) The convergence of the DDT system is analyzed rigorously. It is proven, in the paper, that almost all trajectories of the system starting from points in an invariant set will converge exponentially to the unit eigenvector associated with the largest eigenvalue of the correlation matrix. In addition, exponential convergence rate are obtained, providing useful guidelines for the selection of fast convergence learning rate. 3) Since the trajectories may diverge, the careful choice of initial vectors is an important issue. This paper suggests to use the domain of unit hyper sphere as initial vectors to guarantee convergence. 4) Simulation results will be furnished to illustrate the theoretical results achieved.

In this paper, an observer-based direct adaptive fuzzy-neural control scheme is presented for nonaffine nonlinear systems in the presence of unknown structure of nonlinearities. A direct adaptive fuzzy-neural controller and a class of generalized nonlinear systems, which are called nonaffine nonlinear systems, are instead of the indirect one and affine nonlinear systems given by Leu et al. By using implicit function theorem and Taylor series expansion, the observer-based control law and the weight update law of the fuzzy-neural controller are derived for the nonaffine nonlinear systems. Based on strictly-positive-real (SPR) Lyapunov theory, the stability of the closed-loop system can be verified. Moreover, the overall adaptive scheme guarantees that all signals involved are bounded and the output of the closed-loop system will asymptotically track the desired output trajectory. To demonstrate the effectiveness of the proposed method, simulation results are illustrated in this paper.

Nonnegative matrix factorization (NMF) is an unsupervised learning method useful in various applications including image processing and semantic analysis of documents. This paper focuses on symmetric NMF (SNMF), which is a special case of NMF decomposition. Three parallel multiplicative update algorithms using level 3 basic linear algebra subprograms directly are developed for this problem. First, by minimizing the Euclidean distance, a multiplicative update algorithm is proposed, and its convergence under mild conditions is proved. Based on it, we further propose another two fast parallel methods: α-SNMF and β -SNMF algorithms. All of them are easy to implement. These algorithms are applied to probabilistic clustering. We demonstrate their effectiveness for facial image clustering, document categorization, and pattern clustering in gene expression.

In this paper, we analyze and establish the stability and convergence of the dynamical system proposed by Xia and Feng, whose equilibria solve variational inequality and related problems. Under the pseudo-monotonicity and other conditions, this system is proved to be stable in the sense of Lyapunov and converges to one of its equilibrium points for any starting point. Meanwhile, the global exponential stability of this system is also shown under some mild conditions without the strong monotonicity of the mapping. The obtained results improve and correct some existing ones. The validity and performance of this system are demonstrated by some numerical examples.

In this paper, the direct neural dynamic programming technique is utilized to solve the Hamilton-Jacobi-Bellman equation forward-in-time for the decentralized near optimal regulation of a class of nonlinear interconnected discrete-time systems with unknown internal subsystem and interconnection dynamics, while the input gain matrix is considered known. Even though the unknown interconnection terms are considered weak and functions of the entire state vector, the decentralized control is attempted under the assumption that only the local state vector is measurable. The decentralized nearly optimal controller design for each subsystem consists of two neural networks (NNs), an action NN that is aimed to provide a nearly optimal control signal, and a critic NN which evaluates the performance of the overall system. All NN parameters are tuned online for both the NNs. By using Lyapunov techniques it is shown that all subsystems signals are uniformly ultimately bounded and that the synthesized subsystems inputs approach their corresponding nearly optimal control inputs with bounded error. Simulation results are included to show the effectiveness of the approach.

Spectral clustering (SC) is a large family of grouping methods that partition data using eigenvectors of an affinity matrix derived from the data. Though SC methods have been successfully applied to a large number of challenging clustering scenarios, it is noteworthy that they will fail when there are significant intersections among different clusters. In this paper, based on the analysis that SC methods are able to work well when the affinity values of the points belonging to different clusters are relatively low, we propose a new method, called spectral multi-manifold clustering (SMMC), which is able to handle intersections. In our model, the data are assumed to lie on or close to multiple smooth low-dimensional manifolds, where some data manifolds are separated but some are intersecting. Then, local geometric information of the sampled data is incorporated to construct a suitable affinity matrix. Finally, spectral method is applied to this affinity matrix to group the data. Extensive experiments on synthetic as well as real datasets demonstrate the promising performance of SMMC.

The linear discriminant analysis (LDA) is a very popular linear feature extraction approach. The algorithms of LDA usually perform well under the following two assumptions. The first assumption is that the global data structure is consistent with the local data structure. The second assumption is that the input data classes are Gaussian distributions. However, in real-world applications, these assumptions are not always satisfied. In this paper, we propose an improved LDA framework, the local LDA (LLDA), which can perform well without needing to satisfy the above two assumptions. Our LLDA framework can effectively capture the local structure of samples. According to different types of local data structure, our LLDA framework incorporates several different forms of linear feature extraction approaches, such as the classical LDA and principal component analysis. The proposed framework includes two LLDA algorithms: a vector-based LLDA algorithm and a matrix-based LLDA (MLLDA) algorithm. MLLDA is directly applicable to image recognition, such as face recognition. Our algorithms need to train only a small portion of the whole training set before testing a sample. They are suitable for learning large-scale databases especially when the input data dimensions are very high and can achieve high classification accuracy. Extensive experiments show that the proposed algorithms can obtain good classification results.

This paper presents a new wrapper-based feature selection method for support vector regression (SVR) using its probabilistic predictions. The method computes the importance of a feature by aggregating the difference, over the feature space, of the conditional density functions of the SVR prediction with and without the feature. As the exact computation of this importance measure is expensive, two approximations are proposed. The effectiveness of the measure using these approximations, in comparison to several other existing feature selection methods for SVR, is evaluated on both artificial and real-world problems. The result of the experiments show that the proposed method generally performs better than, or at least as well as, the existing methods, with notable advantage when the dataset is sparse.

In blind source separation, many methods have been proposed to estimate the mixing matrix by exploiting sparsity. However, they often need to know the source number a priori, which is very inconvenient in practice. In this paper, a new method, namely nonlinear projection and column masking (NPCM), is proposed to estimate the mixing matrix. A major advantage of NPCM is that it does not need any knowledge of the source number. In NPCM, the objective function is based on a nonlinear projection and its maxima just correspond to the columns of the mixing matrix. Thus a column can be estimated first by locating a maximum and then deflated by a masking operation. This procedure is repeated until the evaluation of the objective function decreases to zero dramatically. Thus the mixing matrix and the number of sources are estimated simultaneously. Because the masking procedure may result in some small and useless local maxima, particle swarm optimization (PSO) is introduced to optimize the objective function. Feasibility and efficiency of PSO are also discussed. Comparative experimental results show the efficiency of NPCM, especially in the cases where the number of sources is unknown and the sources are relatively less sparse.

This paper proposes a hybrid multilogistic methodology, named logistic regression using initial and radial basis function (RBF) covariates. The process for obtaining the coefficients is carried out in three steps. First, an evolutionary programming (EP) algorithm is applied, in order to produce an RBF neural network (RBFNN) with a reduced number of RBF transformations and the simplest structure possible. Then, the initial attribute space (or, as commonly known as in logistic regression literature, the covariate space) is transformed by adding the nonlinear transformations of the input variables given by the RBFs of the best individual in the final generation. Finally, a maximum likelihood optimization method determines the coefficients associated with a multilogistic regression model built in this augmented covariate space. In this final step, two different multilogistic regression algorithms are applied: one considers all initial and RBF covariates (multilogistic initial-RBF regression) and the other one incrementally constructs the model and applies cross validation, resulting in an automatic covariate selection [simplelogistic initial-RBF regression (SLIRBF)]. Both methods include a regularization parameter, which has been also optimized. The methodology proposed is tested using 18 benchmark classification problems from well-known machine learning problems and two real agronomical problems. The results are compared with the corresponding multilogistic regression methods applied to the initial covariate space, to the RBFNNs obtained by the EP algorithm, and to other probabilistic classifiers, including different RBFNN design methods [e.g., relaxed variable kernel density estimation, support vector machines, a sparse classifier (sparse multinomial logistic regression)] and a procedure similar to SLIRBF but using product unit basis functions. The SLIRBF models are found to be competitive when compared with the corresponding multilogistic regression methods and the RBFEP method. A measure of statistical significance is used, which indicates that SLIRBF reaches the state of the art.

In this paper, we consider the problem of constructing accurate and flexible statistical representations for count data, which we often confront in many areas such as data mining, computer vision, and information retrieval. In particular, we analyze and compare several generative approaches widely used for count data clustering, namely multinomial, multinomial Dirichlet, and multinomial generalized Dirichlet mixture models. Moreover, we propose a clustering approach via a mixture model based on a composition of the Liouville family of distributions, from which we select the Beta-Liouville distribution, and the multinomial. The novel proposed model, which we call multinomial Beta-Liouville mixture, is optimized by deterministic annealing expectation-maximization and minimum description length, and strives to achieve a high accuracy of count data clustering and model selection. An important feature of the multinomial Beta-Liouville mixture is that it has fewer parameters than the recently proposed multinomial generalized Dirichlet mixture. The performance evaluation is conducted through a set of extensive empirical experiments, which concern text and image texture modeling and classification and shape modeling, and highlights the merits of the proposed models and approaches.

The novel contributions of this paper are twofold. First, we demonstrate how to characterize unweighted graphs in a permutation-invariant manner using the polynomial coefficients from the Ihara zeta function, i.e., the Ihara coefficients. Second, we generalize the definition of the Ihara coefficients to edge-weighted graphs. For an unweighted graph, the Ihara zeta function is the reciprocal of a quasi characteristic polynomial of the adjacency matrix of the associated oriented line graph. Since the Ihara zeta function has poles that give rise to infinities, the most convenient numerically stable representation is to work with the coefficients of the quasi characteristic polynomial. Moreover, the polynomial coefficients are invariant to vertex order permutations and also convey information concerning the cycle structure of the graph. To generalize the representation to edge-weighted graphs, we make use of the reduced Bartholdi zeta function. We prove that the computation of the Ihara coefficients for unweighted graphs is a special case of our proposed method for unit edge weights. We also present a spectral analysis of the Ihara coefficients and indicate their advantages over other graph spectral methods. We apply the proposed graph characterization method to capturing graph-class structure and clustering graphs. Experimental results reveal that the Ihara coefficients are more effective than methods based on Laplacian spectra.

This paper presents adaptive neural tracking control for a class of non-affine pure-feedback systems with multiple unknown state time-varying delays. To overcome the design difficulty from non-affine structure of pure-feedback system, mean value theorem is exploited to deduce affine appearance of state variables x(i) as virtual controls α(i), and of the actual control u. The separation technique is introduced to decompose unknown functions of all time-varying delayed states into a series of continuous functions of each delayed state. The novel Lyapunov-Krasovskii functionals are employed to compensate for the unknown functions of current delayed state, which is effectively free from any restriction on unknown time-delay functions and overcomes the circular construction of controller caused by the neural approximation of a function of u and [Formula: see text] . Novel continuous functions are introduced to overcome the design difficulty deduced from the use of one adaptive parameter. To achieve uniformly ultimate boundedness of all the signals in the closed-loop system and tracking performance, control gains are effectively modified as a dynamic form with a class of even function, which makes stability analysis be carried out at the present of multiple time-varying delays. Simulation studies are provided to demonstrate the effectiveness of the proposed scheme.

This paper discusses the problem of what kind of learning model is suitable for the tasks of feature extraction for data representation and suggests two evaluation criteria for nonlinear feature extractors: reconstruction error minimization and similarity preservation. Based on the suggested evaluation criteria, a new type of principal curve-similarity preserving principal curve (SPPC) is proposed. SPPCs minimize the reconstruction error under the condition that the similarity between similar samples are preserved in the extracted features, thus giving researchers effective and reliable cognition of the inner structure of data sets. The existence and properties of SPPCs are analyzed; a practical learning algorithm is proposed and high dimensional extensions of SPPCs are also discussed. Experimental results show the virtues of SPPCs in preserving inner structures of data sets and discovering manifolds with high nonlinearity.

The finite mixture model is widely used in various statistical learning problems. However, the model obtained may contain a large number of components, making it inefficient in practical applications. In this paper, we propose to simplify the mixture model by minimizing an upper bound of the approximation error between the original and the simplified model, under the use of the L (2) distance measure. This is achieved by first grouping similar components together and then performing local fitting through function approximation. The simplified model obtained can then be used as a replacement of the original model to speed up various algorithms involving mixture models during training (e.g., Bayesian filtering, belief propagation) and testing [e.g., kernel density estimation, support vector machine (SVM) testing]. Encouraging results are observed in the experiments on density estimation, clustering-based image segmentation, and simplification of SVM decision functions.

The Adachi neural network (AdNN) is a fascinating neural network (NN) which has been shown to possess chaotic properties, and to also demonstrate associative memory (AM) and pattern recognition (PR) characteristics. Variants of the AdNN have also been used to obtain other PR phenomena, and even blurring. An unsurmountable problem associated with the AdNN and the variants referred to above is that all of them require a quadratic number of computations. This is essentially because the NNs in each case are completely connected graphs. In this paper, we consider how the computations can be significantly reduced by merely using a linear number of computations. To achieves this, we extract from the original completely connected graph one of its spanning trees. We then address the problem of computing the weights for this spanning tree. This is done in such a manner that the modified tree-based NN has approximately the same input-output characteristics, and thus the new weights are themselves calculated using a gradient-based algorithm. By a detailed experimental analysis, we show that the new linear-time AdNN-like network possesses chaotic and PR properties for different settings. As far as we know, such a tree-based AdNN has not been reported, and the results given here are novel.

This paper proposes an uncorrelated multilinear discriminant analysis (UMLDA) framework for the recognition of multidimensional objects, known as tensor objects. Uncorrelated features are desirable in recognition tasks since they contain minimum redundancy and ensure independence of features. The UMLDA aims to extract uncorrelated discriminative features directly from tensorial data through solving a tensor-to-vector projection. The solution consists of sequential iterative processes based on the alternating projection method, and an adaptive regularization procedure is incorporated to enhance the performance in the small sample size (SSS) scenario. A simple nearest-neighbor classifier is employed for classification. Furthermore, exploiting the complementary information from differently initialized and regularized UMLDA recognizers, an aggregation scheme is adopted to combine them at the matching score level, resulting in enhanced generalization performance while alleviating the regularization parameter selection problem. The UMLDA-based recognition algorithm is then empirically shown on face and gait recognition tasks to outperform four multilinear subspace solutions (MPCA, DATER, GTDA, TR1DA) and four linear subspace solutions (Bayesian, LDA, ULDA, R-JD-LDA).

This paper presents a novel recurrent neural network for solving a class of convex quadratic programming (QP) problems, in which the quadratic term in the objective function is the square of the Euclidean norm of the variable. This special structure leads to a set of simple optimality conditions for the problem, based on which the neural network model is formulated. Compared with existing neural networks for general convex QP, the new model is simpler in structure and easier to implement. The new model can be regarded as an improved version of the dual neural network in the literature. Based on the new model, a simple neural network capable of solving the $k$ -winners-take-all ( $k$ -WTA) problem is formulated. The stability and global convergence of the proposed neural network is proved rigorously and substantiated by simulation results.

Model selection in kernel linear discriminant analysis (KLDA) refers to the selection of appropriate parameters of a kernel function and the regularizer. By following the principle of maximum information preservation , this paper formulates the model selection problem as a problem of selecting an optimal kernel-induced space in which different classes are maximally separated from each other. A scatter-matrix-based criterion is developed to measure the “goodness” of a kernel-induced space, and the kernel parameters are tuned by maximizing this criterion. This criterion is computationally efficient and is differentiable with respect to the kernel parameters. Compared with the leave-one-out (LOO) or $k$ -fold cross validation (CV), the proposed approach can achieve a faster model selection, especially when the number of training samples is large or when many kernel parameters need to be tuned. To tune the regularization parameter in the KLDA, our criterion is used together with the method proposed by Saadi (2004). Experiments on benchmark data sets verify the effectiveness of this model selection approach.

In this paper, robust stability problems for interval Cohen-Grossberg neural networks with unknown time-varying delays are investigated. Using linear matrix inequality, M -matrix theory, and Halanay inequality techniques, new sufficient conditions independent of time-varying delays are derived to guarantee the uniqueness and the global robust stability of the equilibrium point of interval Cohen-Grossberg neural networks with time-varying delays. All these results have no restriction on the rate of change of the time-varying delays. Compared to some existing results, these new criteria are less conservative and are more convenient to check. Two numerical examples are used to show the effectiveness of the present results.

The insufficiency of using only second-order statistics and premise of exploiting higher order statistics of the data has been well understood, and more advanced objectives including higher order statistics, especially those stemming from information theory, such as error entropy minimization, are now being studied and applied in many contexts of machine learning and signal processing. In the adaptive system training context, the main drawback of utilizing output error entropy as compared to correlation-estimation-based second-order statistics is the computational load of the entropy estimation, which is usually obtained via a plug-in kernel estimator. Sample-spacing estimates offer computationally inexpensive entropy estimators; however, resulting estimates are not differentiable, hence, not suitable for gradient-based adaptation. In this brief paper, we propose a nonparametric entropy estimator that captures the desirable properties of both approaches. The resulting estimator yields continuously differentiable estimates with a computational complexity at the order of those of the sample-spacing techniques. The proposed estimator is compared with the kernel density estimation (KDE)-based entropy estimator in the supervised neural network training framework with computation time and performance comparisons.

In this paper, a new kernel-based method for data visualization and dimensionality reduction is proposed. A reference point is considered corresponding to additional constraints taken in the problem formulation. In contrast with the class of kernel eigenmap methods, the solution (coordinates in the low-dimensional space) is characterized by a linear system instead of an eigenvalue problem. The kernel maps with a reference point are generated from a least squares support vector machine (LS-SVM) core part that is extended with an additional regularization term for preserving local mutual distances together with reference point constraints. The kernel maps possess primal and dual model representations and provide out-of-sample extensions, e.g., for validation-based tuning. The method is illustrated on toy problems and real-life data sets.