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The support vector machine classifier isUnder the constraints, the two classes lie beyond the parallel support-vector-machine margin boundaries $b+x^T\beta =\pm 1$. Their distance is $2/\Vert \beta {\Vert }_2$, so minimizing the norm maximizes the geometric margin. The observations touching the margin are the support vectors.

A separating hyperplane for signed data satisfies $Y_i({\gamma }_0+X_i^T\beta )>0$ for every $i$; its geometric set is $\{x:{\gamma }_0+x^T\beta =0\}$. The plot marks three support vectors. At the shown fit they lie on the two support-vector-machine margin boundaries, so their signed functional margins satisfyThe solid line is the decision hyperplane $X^{\ast T}\hat{\gamma }=0$, while the dashed parallel lines are $X^{\ast T}\hat{\gamma }=1$ and $X^{\ast T}\hat{\gamma }=-1$.

If $\lambda =0$, every parameter vector with all margins at least one has zero hinge loss. Scaling or changing a separating vector can therefore give another minimizer, so the objective need not select the displayed maximum-margin direction or the same three lines. Positive quadratic regularization selects a finite, minimum-norm compromise.

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 $2/\Vert \hat{\beta }{\Vert }_2$.