Convergence of a generalized version of the modified SMO algorithms given by Keerthi et al. for SVM classifier design is proved. The convergence results are also extended to modified SMO algorithms for solving ν-SVM classifier problems.

Abstract-This paper points out an important source of inefficiency in Smola and Schölkopf's sequential minimal optimization (SMO) algorithm for support vector machine (SVM) regression that is caused by the use of a single threshold value. Using clues from the KKT conditions for the dual problem, two threshold parameters are employed to derive modifications of SMO for regression. These modified algorithms perform significantly faster than the original SMO on the datasets tried.Index Terms-Quadratic programming, regression, sequential minimal optimization (SMO) algorithm, support vector machine (SVM).

Supplementary material is available at the site http://guppy.mpe.nus.edu.sg/~mpessk/SparseLOGREG.shtml

Abstract-In this paper we give a new fast iterative algorithm for support vector machine (SVM) classifier design. The basic problem treated is one that does not allow classification violations. The problem is converted to a problem of computing the nearest point between two convex polytopes. The suitability of two classical nearest point algorithms, due to Gilbert, and Mitchell et al., is studied. Ideas from both these algorithms are combined and modified to derive our fast algorithm. For problems which require classification violations to be allowed, the violations are quadratically penalized and an idea due to Cortes and Vapnik and Frieß is used to convert it to a problem in which there are no classification violations. Comparative computational evaluation of our algorithm against powerful SVM methods such as Platt's sequential minimal optimization shows that our algorithm is very competitive.Index Terms-Classification, nearest point algorithm, quadratic programming, support vector machine.

Progression of hepatocellular carcinoma (HCC) is a stepwise process that proceeds from preneoplastic lesions-including low-grade dysplastic nodules (LGDNs) and high-grade dysplastic nodules (HGDNs)-to advanced HCC. The molecular changes associated with this progression are unclear, however, and the morphological cues thought to distinguish pre-neoplastic lesions from well-differentiated HCC are not universally accepted. To understand the multistep process of hepato-carcinogenesis at the molecular level, we used oligo-nucleotide microarrays to investigate the transcription profiles of 50 hepatocellular nodular lesions ranging from LGDNs to primary HCC (Edmondson grades 1-3). We demonstrated that gene expression profiles can discriminate not only between dysplastic nodules and overt carcinoma but also between different histological grades of HCC via unsupervised hierarchical clustering with 10,376 genes. We identified 3,084 grade-associated genes, correlated with tumor progression, using one-way ANOVA and a one-versus-all unpooled t test. H epatocelluar carcinoma (HCC) is one of the most common malignancies worldwide. The chronic hepatitis resulting from infection with hepatitis B virus or hepatitis C virus and exposure to carcinogens such as aflatoxin B1 are known as major risk factors for HCC. 1 Molecular investigations have recently found that genetic alterations of tumor suppressor genes or oncogenes such as p53, ␤-catenin, and AXIN1 might be involved in the progression to HCC, 2-4 but the frequency of these somatic mutations appears to be low in HCCs. Furthermore, it is unclear how these genetic changes reflect the clinical characteristics of the individual tumors. Therefore, the predominant molecular events underlying HCC in most patients remain unknown.Because HCC typically develops in close association with pre-existing cirrhosis, it is widely believed that a liver with cirrhosis may contain pre-neoplastic nodules that are in an intermediate stage between nonneoplastic regenerating nodules and overtly malignant HCC. 5,6 These nod-