= Kernel support-vector coefficient from hinge activity
For the kernel <support vector machine> objective $n^{-1}\sum_i(1-Y_i(\mu+K_i^T\alpha))_++\lambda\alpha^TK\alpha$, assume $Y_i\in\{-1,1\}$, $\lambda>0$, and that the <kernel matrix> $K$ is invertible. The <subdifferential> optimality equation gives
$$
\alpha_i=Y_it_i/(2n\lambda),\qquad t_i=\begin{cases}1&Y_i(\mu+K_i^T\alpha)<1,\\{}[0,1]&Y_i(\mu+K_i^T\alpha)=1,\\0&Y_i(\mu+K_i^T\alpha)>1.\end{cases}
$$
Thus strict margins greater than one force zero coefficients, while misclassified observations have nonzero coefficients. Invertibility matters: a singular <kernel matrix> allows coefficient changes in its <null space> without changing the fitted <function> or objective.
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