LIBSVM is a library for support vector machines (SVM). Its goal is to help users can easily use SVM as a tool. In this document, we present all its implementation details. 1

Boosting (Freund & Schapire 1996, Schapire & Singer 1998) is one of the most important recent developments in classification methodology. The performance of many classification algorithms can often be dramatically improved by sequentially applying them to reweighted versions of the input data, and taking a weighted majority vote of the sequence of classifiers thereby produced. We show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles, namely additive modeling and maximum likelihood. For the two-class problem, boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum Bernoulli likelihood as a criterion. We develop more direct approximations and show that they exhibit nearly identical results to boosting. Direct multi-class generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multi-class generalizations of boosting in most...

Support vector machines (SVMs) were originally designed for binary classification. How to effectively extend it for multiclass classification is still an ongoing research issue. Several methods have been proposed where typically we construct a multiclass classifier by combining several binary classifiers. Some authors also proposed methods that consider all classes at once. As it is computationally more expensive to solve multiclass problems, comparisons of these methods using large-scale problems have not been seriously conducted. Especially for methods solving multiclass SVM in one step, a much larger optimization problem is required so up to now experiments are limited to small data sets. In this paper we give decomposition implementations for two such “all-together” methods. We then compare their performance with three methods based on binary classifications: “one-against-all,” “one-against-one,” and directed acyclic graph SVM (DAGSVM). Our experiments indicate that the “one-against-one” and DAG methods are more suitable for practical use than the other methods. Results also show that for large problems methods by considering all data at once in general need fewer support vectors.

We discuss a strategy for polychotomous classification that involves estimating class probabilities for each pair of classes, and then coupling the estimates together. The coupling model is similar to the Bradley-Terry method for paired comparisons. We study the nature of the class probability estimates that arise, and examine the performance of the procedure in real and simulated datasets. Classifiers used include linear discriminants, nearest neighbors, adaptive nonlinear methods, and the support vector machine. Department of Statistics, Sequoia Hall, Stanford University, Stanford California 94305; trevor@playfair.stanford.edu y Department of Preventive Medicine and Biostatistics, and Department of Statistics; tibs@utstat.toronto.edu 1 Introduction We consider the discrimination problem with K classes and N training observations. The training observations consist of predictor measurements x = (x 1 ; x 2 ; : : : x p ) on p predictors and the known class memberships. Our goal is...

We present a new learning architecture: the Decision Directed Acyclic Graph (DDAG), which is used to combine many two-class classifiers into a multiclass classifier. For an-class problem, the DDAG contains � classifiers, one for each pair of classes. We present a VC analysis of the case when the node classifiers are hyperplanes; the resulting bound on the test error depends on and on the margin achieved at the nodes, but not on the dimension of the space. This motivates an algorithm, DAGSVM, which operates in a kernel-induced feature space and uses two-class maximal margin hyperplanes at each decision-node of the DDAG. The DAGSVM is substantially faster to train and evaluate than either the standard algorithm or Max Wins, while maintaining comparable accuracy to both of these algorithms. 1

Two-category support vector machines (SVM) have been very popular in the machine learning community for classi � cation problems. Solving multicategory problems by a series of binary classi � ers is quite common in the SVM paradigm; however, this approach may fail under various circumstances. We propose the multicategory support vector machine (MSVM), which extends the binary SVM to the multicategory case and has good theoretical properties. The proposed method provides a unifying framework when there are either equal or unequal misclassi � cation costs. As a tuning criterion for the MSVM, an approximate leave-one-out cross-validation function, called Generalized Approximate Cross Validation, is derived, analogous to the binary case. The effectiveness of the MSVM is demonstrated through the applications to cancer classi � cation using microarray data and cloud classi � cation with satellite radiance pro � les.

Pairwise coupling is a popular multi-class classification method that combines together all pairwise comparisons for each pair of classes. This paper presents two approaches for obtaining class probabilities. Both methods can be reduced to linear systems and are easy to implement.

We propose a content-based soft annotation (CBSA) procedure for providing images with semantical labels. The annotation procedure starts with labeling a small set of training images, each with one single semantical label (e.g., forest, animal, or sky). An ensemble of binary classifiers is then trained for predicting label membership for images. The trained ensemble is applied to each individual image to give the image multiple soft labels, and each label is associated with a label membership factor. To select a base binary-classifier for CBSA, we experiment with two learning methods, Support Vector Machines (SVMs) and Bayes Point Machines (BPMs, and compare their class-prediction accuracy. Our empirical study on a 116-category 25K-image set shows that the BPM-based ensemble provides better annotation quality than the SVM-based ensemble for supporting multimodal image retrievals. Keywords: Bayes Point Machines, Support Vector Machines, image annotation, multimodal image retrieval.

2 Everything Old Is New Again: A Fresh Look at Historical

In this paper, we discuss round robin classification (aka pairwise classification), a technique for handling multi-class problems with binary classifiers by learning one classifier for each pair of classes. We present an empirical evaluation of the method, implemented as a wrapper around the Ripper rule learning algorithm, on 20 multi-class datasets from the UCI database repository. Our results show that the technique is very likely to improve Ripper's classification accuracy without having a high risk of decreasing it. More importantly, we give a general theoretical analysis of the complexity of the approach and show that its run-time complexity is below that of the commonly used one-against-all technique.