## Hybrid Misclassification Minimization (1995)

Venue: | Advances in Computational Mathematics |

Citations: | 14 - 7 self |

### BibTeX

@INPROCEEDINGS{Chen95hybridmisclassification,

author = {Chunhui Chen and O. L. Mangasarian},

title = {Hybrid Misclassification Minimization},

booktitle = {Advances in Computational Mathematics},

year = {1995},

pages = {127--136}

}

### OpenURL

### Abstract

Given two finite point sets A and B in the n-dimensional real space R n , we consider the NP-complete problem of minimizing the number of misclassified points by a plane attempting to divide R n into two halfspaces such that each open halfspace contains points mostly of A or B . This problem is equivalent to determining a plane fx j x T w = flg that maximizes the number of points x 2 A satisfying x T w ? fl, plus the number of points x 2 B satisfying x T w ! fl. A simple but fast algorithm is proposed that alternates between (i) minimizing the number of misclassified points by translation of the separating plane, and (ii) a rotation of the plane so that it minimizes a weighted average sum of the distances of the misclassified points to the separating plane. Existence of a global solution to an underlying hybrid minimization problem is established. Computational comparison with a parametric approach to solve the NP-complete problem indicates that our approach is considerably ...

### Citations

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Citation Context ...ter. 3 Numerical Computation and Comparisons We report now on numerical results on the Wisconsin Breast Cancer Database (WBCD) and other data sets from the Irvine Machine Learning Database Repository =-=[10]-=- as well as the Star/Galaxy database collected by Odewahn [12] and the Wisconsin Breast Cancer Prognosis Database [9, 16]. For each data set, a separating plane was obtained by three methods: the para... |

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Citation Context ... present approach we shall propose a fast alternative hybrid criterion that is quite effective in approximately minimizing the number of misclassified points as determined by tenfold cross-validation =-=[15]-=-. The basic idea is to minimize the number of misclassified points by translating the separating plane, and then rotating the plane in order to minimize a weighted average sum of the distances of misc... |

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Citation Context ... point. The idea of using different criteria to determine different parts of the solution (w; fl) is similar to that of finding equilibrium points [13] and solving multicriteria optimization problems =-=[14]-=-. Before defining precisely our problem, we slightly modify the misclassification minimization problem (6) as follows: min w;fl e T (e \Gamma (Aw \Gamma efl)s) + e T (e \Gamma (\GammaBw + efl)s) (7) W... |

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Citation Context ...MM average of 2.32 PMM average of 22.3 Table 1. Comparison of Hybrid Misclassification Minimization (HMM) with Parametric Misclassification Minimization (PMM) [8, 1] & Robust Linear Programming (RLP) =-=[2]-=- m Training Set Correctness Date Set k Testing Set Correctness n Time Seconds SPARCstation 20 Average LPs Solved HMM PMM RLP 28 89.12 95.92 84.343 WBC Prognosis 119 72.24 71.33 66.048 32 0.71 10.65 0.... |

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Citation Context ...ed was the average time for the ten different subsets used for training. The parametric misclassification minimization procedure was coded in the modeling language AMPL [3] in [1] utilizing the MINOS =-=[11]-=- linear programming solver. The HMM Algorithm and the robust linear program algorithm were implemented using C and called MINOS as a subroutine to solve the linear programs. Table 1 gives a summary of... |

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Citation Context ...nts is possible. We term such a point as a stationary point. The idea of using different criteria to determine different parts of the solution (w; fl) is similar to that of finding equilibrium points =-=[13]-=- and solving multicriteria optimization problems [14]. Before defining precisely our problem, we slightly modify the misclassification minimization problem (6) as follows: min w;fl e T (e \Gamma (Aw \... |

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Citation Context ...fic Research Grant F49620-94-1-0036 and National Science Foundation Grant CCR-9322479. plane (1) that minimizes a weighted average of the sum of the distances of the misclassified points to the plane =-=[7, 2]-=- as follows: min w;fl;y;z ( e T y m + e T z k j Aw + ysefl + e; Bw \Gamma zsefl \Gamma e; ys0; zs0 ) (3) Here the rows of the matrices A 2 R m\Thetan and B 2 R k\Thetan represent the m points in A and... |

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Citation Context ...isclassification minimization problem min w;fl e T (\GammaAw + efl + e)s+ e T (Bw \Gamma efl + e)s: (6) In [8, 1] this problem was reformulated as a linear program with equilibrium constraints (LPEC) =-=[6]-=-, that is a linear program with a single complementarity constraint. An implicitly exact penalty method as well as a parametric method were proposed for solving the LPEC in [8] and successfully implem... |

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Citation Context ...y more difficult and in fact is NP-complete, as shown in Proposition 2 of Section 2 below. This problem was considered in [8], where a parametric minimization approach was proposed and implemented in =-=[1]-=-. Although the parametric procedure is effective, it is costly computationally, which is to be expected since the underlying problem is NP-complete. In the present approach we shall propose a fast alt... |

8 |
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Citation Context ...e (WBCD) and other data sets from the Irvine Machine Learning Database Repository [10] as well as the Star/Galaxy database collected by Odewahn [12] and the Wisconsin Breast Cancer Prognosis Database =-=[9, 16]-=-. For each data set, a separating plane was obtained by three methods: the parametric misclassification minimization (PMM) procedure of [8, 1], the HMM Algorithm 5 of Section 2, and the robust linear ... |