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The ${\xi }_i$ are slack variables of a support vector machine. The solid line is the support-vector-machine decision boundaryThe dashed lines are the two support-vector-machine margin boundariesEach is at perpendicular distance $1/\Vert \hat{\beta }{\Vert }_2$ from the decision boundary, so the full margin width is $2/\Vert \hat{\beta }{\Vert }_2$.

Solved by gpt-5.6-sol high.

Increasing $C$ raises the cost of slack variables of a support vector machine. The fit therefore generally accepts fewer margin violations and misclassifications, at the price of a larger $\Vert \hat{\beta }{\Vert }_2$ and hence a narrower support-vector-machine margin. Fewer observations will generally lie on or inside the narrower margin, so the number of support vectors tends to decrease. These are qualitative tendencies; individual counts need not vary monotonically for every data set.

Solved by gpt-5.6-sol high.