Feasibility and Finite Convergence Analysis for Accurate On-Line $\nu$-Support Vector Machine

The ν-support vector machine ( ν-SVM) for classification has the advantage of using a parameter ν on controlling the number of support vectors and margin errors. Recently, an interesting accurate on-line algorithm accurate on-line ν-SVM algorithm (AONSVM) is proposed for training ν-SVM. AONSVM can be viewed as a special case of parametric quadratic programming techniques. It is demonstrated that AONSVM avoids the infeasible updating path as far as possible, and successfully converges to the optimal solution based on experimental analysis. However, because of the differences between AONSVM and classical parametric quadratic programming techniques, there is no theoretical justification for these conclusions. In this paper, we prove the feasibility and finite convergence of AONSVM under two assumptions. The main results of feasibility analysis include: 1) the inverses of the two key matrices in AONSVM always exist; 2) the rules for updating the two key inverse matrices are reliable; 3) the variable ζ can control the adjustment of the sum of all the weights efficiently; and 4) a sample cannot migrate back and forth in successive adjustment steps among the set of margin support vectors, the set of error support vectors, and the set of the remaining vectors. Moreover, the analyses of AONSVM also provide the proofs of the feasibility and finite convergence for accurate on-line C-SVM learning directly.