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i.e. we find that for the offsetb=aTw, which is the projection ofaonto to the vectorw. Without loss of generality we may thus chooseaperpendicular to the plane, in which case the length||a|| = |b|/||w||represents the shortest, orthogonal distance between the origin and the hyperplane.

We now define 2 more hyperplanes parallel to the separating hyperplane. They represent that planes that cut through the closest training examples on either side. We will call them “support hyper-planes” in the following, because the datavectors they contain support the plane.

We define the distance between the these hyperplanes and the separating hyperplane to bed+andd−respectively. Themargin,γ, is defined to bed++d−.Our goal is now to find a the separating hyperplane so that the margin is largest, while the separating hyperplane is equidistant from both. We can write the following equations for the support hyperplanes: