Support vector ordinal regression (SVOR) is a popular method to tackle ordinal regression problems. However, until now there were no effective algorithms proposed to address incremental SVOR learning due to the complicated formulations of SVOR. Recently, an interesting accurate on-line algorithm was proposed for training ν -support vector classification (ν-SVC), which can handle a quadratic formulation with a pair of equality constraints. In this paper, we first present a modified SVOR formulation based on a sum-of-margins strategy. The formulation has multiple constraints, and each constraint includes a mixture of an equality and an inequality. Then, we extend the accurate on-line ν-SVC algorithm to the modified formulation, and propose an effective incremental SVOR algorithm. The algorithm can handle a quadratic formulation with multiple constraints, where each constraint is constituted of an equality and an inequality. More importantly, it tackles the conflicts between the equality and inequality constraints. We also provide the finite convergence analysis for the algorithm. Numerical experiments on the several benchmark and real-world data sets show that the incremental algorithm can converge to the optimal solution in a finite number of steps, and is faster than the existing batch and incremental SVOR algorithms. Meanwhile, the modified formulation has better accuracy than the existing incremental SVOR algorithm, and is as accurate as the sum-of-margins based formulation of Shashua and Levin.

The ν -support vector classification has the advantage of using a regularization parameter ν to control the number of support vectors and margin errors. Recently, a regularization path algorithm for ν -support vector classification ( ν -SvcPath) suffers exceptions and singularities in some special cases. In this brief, we first present a new equivalent dual formulation for ν -SVC and, then, propose a robust ν -SvcPath, based on lower upper decomposition with partial pivoting. Theoretical analysis and experimental results verify that our proposed robust regularization path algorithm can avoid the exceptions completely, handle the singularities in the key matrix, and fit the entire solution path in a finite number of steps. Experimental results also show that our proposed algorithm fits the entire solution path with fewer steps and less running time than original one does.

Minimax probability machine (MPM) is an interesting discriminative classifier based on generative prior knowledge. It can directly estimate the probabilistic accuracy bound by minimizing the maximum probability of misclassification. The structural information of data is an effective way to represent prior knowledge, and has been found to be vital for designing classifiers in real-world problems. However, MPM only considers the prior probability distribution of each class with a given mean and covariance matrix, which does not efficiently exploit the structural information of data. In this paper, we use two finite mixture models to capture the structural information of the data from binary classification. For each subdistribution in a finite mixture model, only its mean and covariance matrix are assumed to be known. Based on the finite mixture models, we propose a structural MPM (SMPM). SMPM can be solved effectively by a sequence of the second-order cone programming problems. Moreover, we extend a linear model of SMPM to a nonlinear model by exploiting kernelization techniques. We also show that the SMPM can be interpreted as a large margin classifier and can be transformed to support vector machine and maxi-min margin machine under certain special conditions. Experimental results on both synthetic and real-world data sets demonstrate the effectiveness of SMPM.

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