Support vector (${\alpha }_i>0$)

In a support vector machine, a dual support vector is a training observation with positive dual coefficient ${\alpha }_i>0$. Such an observation appears in the fitted score $f(x)={\sum }_i{\alpha }_i{y}_{i}k(x_i,x)+b$, where $k$ is the positive-definite kernel.

For a soft-margin support vector machine, the Karush-Kuhn-Tucker conditions imply that every dual support vector has signed margin $y_{i}f(x_i)\le 1$. The converse can fail at equality: a point exactly on a margin boundary may have ${\alpha }_i=0$ in a degenerate solution. Points strictly inside the margin have ${\alpha }_i=C$ in the usual linear-slack formulation. Some texts instead use a geometric convention counting all observations on or inside the margin. The dual support vectors and margin degeneracy distinction matters for support-vector leave-one-out error bounds.