Adaptive Learning for Robust Radial Basis Function Networks

This article addresses the robust estimation of the output layer linear parameters in a radial basis function network (RBFN). A prominent method used to estimate the output layer parameters in an RBFN with the predetermined hidden layer parameters is the least-squares estimation, which is the maximum-likelihood (ML) solution in the specific case of the Gaussian noise. We highlight the connection between the ML estimation and minimizing the Kullback-Leibler (KL) divergence between the actual noise distribution and the assumed Gaussian noise. Based on this connection, a method is proposed using a variant of a generalized KL divergence, which is known to be more robust to outliers in the pattern recognition and machine-learning problems. The proposed approach produces a surrogate-likelihood function, which is robust in the sense that it is adaptive to a broader class of noise distributions. Several signal processing experiments are conducted using artificially generated and real-world data. It is shown that in all cases, the proposed adaptive learning algorithm outperforms the standard approaches in terms of mean-squared error (MSE). Using the relative increase in the MSE for different noise conditions, we compare the robustness of our proposed algorithm with the existing methods for robust RBFN training and show that our method results in overall improvement in terms of absolute MSE values and consistency.

Multicolumn RBF Network

This paper proposes the multicolumn RBF network (MCRN) as a method to improve the accuracy and speed of a traditional radial basis function network (RBFN). The RBFN, as a fully connected artificial neural network (ANN), suffers from costly kernel inner-product calculations due to the use of many instances as the centers of hidden units. This issue is not critical for small datasets, as adding more hidden units will not burden the computation time. However, for larger datasets, the RBFN requires many hidden units with several kernel computations to generalize the problem. The MCRN mechanism is constructed based on dividing a dataset into smaller subsets using the k-d tree algorithm. resultant subsets are considered as separate training datasets to train individual RBFNs. Those small RBFNs are stacked in parallel and bulged into the MCRN structure during testing. The MCRN is considered as a well-developed and easy-to-use parallel structure, because each individual ANN has been trained on its own subsets and is completely separate from the other ANNs. This parallelized structure reduces the testing time compared with that of a single but larger RBFN, which cannot be easily parallelized due to its fully connected structure. Small informative subsets provide the MCRN with a regional experience to specify the problem instead of generalizing it. The MCRN has been tested on many benchmark datasets and has shown better accuracy and great improvements in training and testing times compared with a single RBFN. The MCRN also shows good results compared with those of some machine learning techniques, such as the support vector machine and k-nearest neighbors.

Learning Subspace-Based RBFNN Using Coevolutionary Algorithm for Complex Classification Tasks

Many real-world classification problems are characterized by samples of a complex distribution in the input space. The classification accuracy is determined by intrinsic properties of all samples in subspaces of features. This paper proposes a novel algorithm for the construction of radial basis function neural network (RBFNN) classifier based on subspace learning. In this paper, feature subspaces are obtained for every hidden node of the RBFNN during the learning process. The connection weights between the input layer and the hidden layer are adjusted to produce various subspaces with dominative features for different hidden nodes. The network structure and dominative features are encoded in two subpopulations that are cooperatively coevolved using the coevolutionary algorithm to achieve a better global optimality for the estimated RBFNN. Experimental results illustrate that the proposed algorithm is able to obtain RBFNN models with both better classification accuracy and simpler network structure when compared with other learning algorithms. Thus, the proposed model provides a more flexible and efficient approach to complex classification tasks by employing the local characteristics of samples in subspaces.

A growing and pruning sequential learning algorithm of hyper basis function neural network for function approximation

Radial basis function (RBF) neural network is constructed of certain number of RBF neurons, and these networks are among the most used neural networks for modeling of various nonlinear problems in engineering. Conventional RBF neuron is usually based on Gaussian type of activation function with single width for each activation function. This feature restricts neuron performance for modeling the complex nonlinear problems. To accommodate limitation of a single scale, this paper presents neural network with similar but yet different activation function-hyper basis function (HBF). The HBF allows different scaling of input dimensions to provide better generalization property when dealing with complex nonlinear problems in engineering practice. The HBF is based on generalization of Gaussian type of neuron that applies Mahalanobis-like distance as a distance metrics between input training sample and prototype vector. Compared to the RBF, the HBF neuron has more parameters to optimize, but HBF neural network needs less number of HBF neurons to memorize relationship between input and output sets in order to achieve good generalization property. However, recent research results of HBF neural network performance have shown that optimal way of constructing this type of neural network is needed; this paper addresses this issue and modifies sequential learning algorithm for HBF neural network that exploits the concept of neuron's significance and allows growing and pruning of HBF neuron during learning process. Extensive experimental study shows that HBF neural network, trained with developed learning algorithm, achieves lower prediction error and more compact neural network.

Nonlinear regularization path for quadratic loss support vector machines

Regularization path algorithms have been proposed to deal with model selection problem in several machine learning approaches. These algorithms allow computation of the entire path of solutions for every value of regularization parameter using the fact that their solution paths have piecewise linear form. In this paper, we extend the applicability of regularization path algorithm to a class of learning machines that have quadratic loss and quadratic penalty term. This class contains several important learning machines such as squared hinge loss support vector machine (SVM) and modified Huber loss SVM. We first show that the solution paths of this class of learning machines have piecewise nonlinear form, and piecewise segments between two breakpoints are characterized by a class of rational functions. Then we develop an algorithm that can efficiently follow the piecewise nonlinear path by solving these rational equations. To solve these rational equations, we use rational approximation technique with quadratic convergence rate, and thus, our algorithm can follow the nonlinear path much more precisely than existing approaches such as predictor-corrector type nonlinear-path approximation. We show the algorithm performance on some artificial and real data sets.