M3G: Maximum Margin Microarray Gridding

margin between spots of distinct rows or columns. The width of the margin is equal to 2/�w�, therefore the widest margin is found by minimizing �w� under the constraints ci(w̄ · x̄i − b) ≥ 1, i.e. requiring that all the spots lie on the correct side of the resulting grid line. The support vector machine described above is called a hard-margin SVM and does not take into account any outliers. One of its properties is that the separating line is solely determined by the vectors that that lie on the edges of the margin, called support vectors. In a linear SVM, a very small number of support vectors determine the separating line and the margin. In the case of outliers present inside the margin, the positioning of the separating line will be exclusively determined by the outliers and will thus be suboptimal for gridding. This problem can be solved using the soft-margin SVM, where a slack variable ξi is introduced for each vector x̄i. The constraints are then formulated as ci(w̄ · x̄i− b) ≥ 1− ξi and the separating hyperplane can be found by minimizing