On the consistency of multiclass classification methods

Binary classification methods can be generalized in many ways to handle multiple classes. It turns out that not all generalizations preserve the nice property of Bayes consistency. We provide a necessary and sufficient condition for consistency which applies to a large class of multiclass classification methods. The approach is illustrated by applying it to some multiclass methods proposed in the literature.

On the consistency of multiclass classification methods

Binary classification is a well studied special case of the classification problem. Statistical properties of binary classifiers, such as consistency, have been investigated in a variety of settings. Binary classification methods can be generalized in many ways to handle multiple classes. It turns out that one can lose consistency in generalizing a binary classification method to deal with multiple classes. We study a rich family of multiclass methods and provide a necessary and sufficient condition for their consistency. We illustrate our approach by applying it to some multiclass methods proposed in the literature.

Generative Maximum Entropy Learning for Multiclass Classification

Maximum entropy approach to classification is very well studied in applied statistics and machine learning and almost all the methods that exists in literature are discriminative in nature. In this paper, we introduce a maximum entropy classification method with feature selection for large dimensional data such as text datasets that is generative in nature. To tackle the curse of dimensionality of large data sets, we employ conditional independence assumption (Naive Bayes) and we perform feature selection simultaneously, by enforcing a `maximum discrimination' between estimated class conditional densities. For two class problems, in the proposed method, we use Jeffreys (J) divergence to discriminate the class conditional densities. To extend our method to the multi-class case, we propose a completely new approach by considering a multi-distribution divergence: we replace Jeffreys divergence by Jensen-Shannon (JS) divergence to discriminate conditional densities of multiple classes. In order to reduce computational complexity, we employ a modified Jensen-Shannon divergence (JS(GM)), based on AM-GM inequality. We show that the resulting divergence is a natural generalization of Jeffreys divergence to a multiple distributions case. As far as the theoretical justifications are concerned we show that when one intends to select the best features in a generative maximum entropy approach, maximum discrimination using J-divergence emerges naturally in binary classification. Performance and comparative study of the proposed algorithms have been demonstrated on large dimensional text and gene expression datasets that show our methods scale up very well with large dimensional datasets.

Multiclass Classification of SRBCTs

A novel approach to multiclass tumor classification using Artificial Neural Networks (ANNs) was introduced in a recent paper cite{Khan2001}. The method successfully classified and diagnosed small, round blue cell tumors (SRBCTs) of childhood into four distinct categories, neuroblastoma (NB), rhabdomyosarcoma (RMS), non-Hodgkin lymphoma (NHL) and the Ewing family of tumors (EWS), using cDNA gene expression profiles of samples that included both tumor biopsy material and cell lines. We report that using an approach similar to the one reported by Yeang et al cite{Yeang2001}, i.e. multiclass classification by combining outputs of binary classifiers, we achieved equal accuracy with much fewer features. We report the performances of 3 binary classifiers (k-nearest neighbors (kNN), weighted-voting (WV), and support vector machines (SVM)) with 3 feature selection techniques (Golub's Signal to Noise (SN) ratios cite{Golub99}, Fisher scores (FSc) and Mukherjee's SVM feature selection (SVMFS))cite{Sayan98}.

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.

Using the EM Algorithm to Train Neural Networks: Misconceptions and a New Algorithm for Multiclass Classification

The expectation-maximization (EM) algorithm has been of considerable interest in recent years as the basis for various algorithms in application areas of neural networks such as pattern recognition. However, there exists some misconceptions concerning its application to neural networks. In this paper, we clarify these misconceptions and consider how the EM algorithm can be adopted to train multilayer perceptron (MLP) and mixture of experts (ME) networks in applications to multiclass classification. We identify some situations where the application of the EM algorithm to train MLP networks may be of limited value and discuss some ways of handling the difficulties. For ME networks, it is reported in the literature that networks trained by the EM algorithm using iteratively reweighted least squares (IRLS) algorithm in the inner loop of the M-step, often performed poorly in multiclass classification. However, we found that the convergence of the IRLS algorithm is stable and that the log likelihood is monotonic increasing when a learning rate smaller than one is adopted. Also, we propose the use of an expectation-conditional maximization (ECM) algorithm to train ME networks. Its performance is demonstrated to be superior to the IRLS algorithm on some simulated and real data sets.

Comparison of Multiclass SVM Classification Methods to Use in a Supportive System for Distance Relay Coordination

This paper aims at evaluating the methods of multiclass support vector machines (SVMs) for effective use in distance relay coordination. Also, it describes a strategy of supportive systems to aid the conventional protection philosophy in combating situations where protection systems have maloperated and/or information is missing and provide selective and secure coordinations. SVMs have considerable potential as zone classifiers of distance relay coordination. This typically requires a multiclass SVM classifier to effectively analyze/build the underlying concept between reach of different zones and the apparent impedance trajectory during fault. Several methods have been proposed for multiclass classification where typically several binary SVM classifiers are combined together. Some authors have extended binary SVM classification to one-step single optimization operation considering all classes at once. In this paper, one-step multiclass classification, one-against-all, and one-against-one multiclass methods are compared for their performance with respect to accuracy, number of iterations, number of support vectors, training, and testing time. The performance analysis of these three methods is presented on three data sets belonging to training and testing patterns of three supportive systems for a region and part of a network, which is an equivalent 526-bus system of the practical Indian Western grid.

Classification calibration dimension for general multiclass losses

We study consistency properties of surrogate loss functions for general multiclass classification problems, defined by a general loss matrix. We extend the notion of classification calibration, which has been studied for binary and multiclass 0-1 classification problems (and for certain other specific learning problems), to the general multiclass setting, and derive necessary and sufficient conditions for a surrogate loss to be classification calibrated with respect to a loss matrix in this setting. We then introduce the notion of \emph{classification calibration dimension} of a multiclass loss matrix, which measures the smallest `size' of a prediction space for which it is possible to design a convex surrogate that is classification calibrated with respect to the loss matrix. We derive both upper and lower bounds on this quantity, and use these results to analyze various loss matrices. In particular, as one application, we provide a different route from the recent result of Duchi et al.\ (2010) for analyzing the difficulty of designing `low-dimensional' convex surrogates that are consistent with respect to pairwise subset ranking losses. We anticipate the classification calibration dimension may prove to be a useful tool in the study and design of surrogate losses for general multiclass learning problems.

Improving Multiclass Text Classification with the Support Vector Machine

We compare Naive Bayes and Support Vector Machines on the task of multiclass text classification. Using a variety of approaches to combine the underlying binary classifiers, we find that SVMs substantially outperform Naive Bayes. We present full multiclass results on two well-known text data sets, including the lowest error to date on both data sets. We develop a new indicator of binary performance to show that the SVM's lower multiclass error is a result of its improved binary performance. Furthermore, we demonstrate and explore the surprising result that one-vs-all classification performs favorably compared to other approaches even though it has no error-correcting properties.

Multiclass Core Vector Machine

Even though several techniques have been proposed in the literature for achieving multiclass classification using Support Vector Machine(SVM), the scalability aspect of these approaches to handle large data sets still needs much of exploration. Core Vector Machine(CVM) is a technique for scaling up a two class SVM to handle large data sets. In this paper we propose a Multiclass Core Vector Machine(MCVM). Here we formulate the multiclass SVM problem as a Quadratic Programming(QP) problem defining an SVM with vector valued output. This QP problem is then solved using the CVM technique to achieve scalability to handle large data sets. Experiments done with several large synthetic and real world data sets show that the proposed MCVM technique gives good generalization performance as that of SVM at a much lesser computational expense. Further, it is observed that MCVM scales well with the size of the data set.

Improving Multi-class Text Classification with Naive Bayes

There are numerous text documents available in electronic form. More and more are becoming available every day. Such documents represent a massive amount of information that is easily accessible. Seeking value in this huge collection requires organization; much of the work of organizing documents can be automated through text classification. The accuracy and our understanding of such systems greatly influences their usefulness. In this paper, we seek 1) to advance the understanding of commonly used text classification techniques, and 2) through that understanding, improve the tools that are available for text classification. We begin by clarifying the assumptions made in the derivation of Naive Bayes, noting basic properties and proposing ways for its extension and improvement. Next, we investigate the quality of Naive Bayes parameter estimates and their impact on classification. Our analysis leads to a theorem which gives an explanation for the improvements that can be found in multiclass classification with Naive Bayes using Error-Correcting Output Codes. We use experimental evidence on two commonly-used data sets to exhibit an application of the theorem. Finally, we show fundamental flaws in a commonly-used feature selection algorithm and develop a statistics-based framework for text feature selection. Greater understanding of Naive Bayes and the properties of text allows us to make better use of it in text classification.

Building binary-tree-based multiclass classifiers using separability measures

Various popular machine learning techniques, like support vector machines, are originally conceived for the solution of two-class (binary) classification problems. However, a large number of real problems present more than two classes. A common approach to generalize binary learning techniques to solve problems with more than two classes, also known as multiclass classification problems, consists of hierarchically decomposing the multiclass problem into multiple binary sub-problems, whose outputs are combined to define the predicted class. This strategy results in a tree of binary classifiers, where each internal node corresponds to a binary classifier distinguishing two groups of classes and the leaf nodes correspond to the problem classes. This paper investigates how measures of the separability between classes can be employed in the construction of binary-tree-based multiclass classifiers, adapting the decompositions performed to each particular multiclass problem. (C) 2010 Elsevier B.V. All rights reserved.

A review on the combination of binary classifiers in multiclass problems

Several real problems involve the classification of data into categories or classes. Given a data set containing data whose classes are known, Machine Learning algorithms can be employed for the induction of a classifier able to predict the class of new data from the same domain, performing the desired discrimination. Some learning techniques are originally conceived for the solution of problems with only two classes, also named binary classification problems. However, many problems require the discrimination of examples into more than two categories or classes. This paper presents a survey on the main strategies for the generalization of binary classifiers to problems with more than two classes, known as multiclass classification problems. The focus is on strategies that decompose the original multiclass problem into multiple binary subtasks, whose outputs are combined to obtain the final prediction.

Evolutionary tuning of SVM parameter values in multiclass problems

Support vector machines (SVMs) were originally formulated for the solution of binary classification problems. In multiclass problems, a decomposition approach is often employed, in which the multiclass problem is divided into multiple binary subproblems, whose results are combined. Generally, the performance of SVM classifiers is affected by the selection of values for their parameters. This paper investigates the use of genetic algorithms (GAs) to tune the parameters of the binary SVMs in common multiclass decompositions. The developed GA may search for a set of parameter values common to all binary classifiers or for differentiated values for each binary classifier. (C) 2008 Elsevier B.V. All rights reserved.

HIERARCHICAL DECOMPOSITION OF MULTICLASS PROBLEMS

Several popular Machine Learning techniques are originally designed for the solution of two-class problems. However, several classification problems have more than two classes. One approach to deal with multiclass problems using binary classifiers is to decompose the multiclass problem into multiple binary sub-problems disposed in a binary tree. This approach requires a binary partition of the classes for each node of the tree, which defines the tree structure. This paper presents two algorithms to determine the tree structure taking into account information collected from the used dataset. This approach allows the tree structure to be determined automatically for any multiclass dataset.