Support vector machine

A support vector machine chooses a separating decision function by trading a wide geometric margin against hinge-loss violations.

A kernel support vector machine trains a support vector machine using a positive-definite kernel instead of explicit feature coordinates. The kernel trick evaluates all required feature inner products through the training Gram matrix; prediction is ${\sum }_i{\alpha }_i{y}_{i}k(x_i,x)+b$. Nonzero dual coefficients identify support vectors. An unpenalized intercept supplies the dual equality ${\sum }_i{\alpha }_i{y}_i=0$. The Reproducing kernel Hilbert space construction explains why a positive-semidefinite kernel defines valid feature geometry.

For signed training labels, a linear soft-margin support vector machine minimizeswith $C>0$. Its Karush-Kuhn-Tucker conditions give $w={\sum }_i{\alpha }_i{y}_i{x}_i$, ${\sum }_i{\alpha }_i{y}_i=0$, and $0\le {\alpha }_i\le C$.

For the kernel support vector machine objective $n^{-1}{\sum }_i(1-Y_i(\mu +K_i^T\alpha ){)}_{+}+\lambda {\alpha }^{T}K\alpha$, assume $Y_i\in \{-1,1\}$, $\lambda >0$, and that the kernel matrix $K$ is invertible. The subdifferential optimality equation givesThus 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.

For a fixed $C$ in the unnormalized sum-of-slacks objective and unique training optimizers, deleting an observation with ${\alpha }_i=0$ preserves the Karush-Kuhn-Tucker conditions and the fitted decision function. It is correctly classified when held out. Consequently Leave-one-out cross-validation makes at most $s$ errors, where $s$ counts positive dual coefficients. With geometric support vectors, counting all points of signed margin at most one also gives the bound, without a nondegeneracy assumption.

A dual support vector has ${\alpha }_i>0$. Every such point has signed margin at most one, but a point exactly on the margin can have ${\alpha }_i=0$ in a degenerate solution. If ${\alpha }_i=0$, complementary slackness forces ${\xi }_i=0$, so the point is correctly classified with margin at least one.

For signed observations $(x_i,y_i)$, an affine hyperplane with score $f(x)=\alpha +x^T\beta$ is separating when $y_{i}f(x_i)>0$ for every observation.

The slack variables satisfy ${\xi }_i\ge \max \{0,1-y_{i}f(x_i)\}$ and quantify margin violations.

For a linear score $f(x)=\alpha +x^T\beta$, the decision boundary is $f(x)=0$.

For a linear support vector machine, the geometric margin lies between $f(x)=-1$ and $f(x)=1$ and has width $2/\Vert \beta \Vert$.

The two margin boundaries of a linear support vector machine are the parallel hyperplanes $f(x)=1$ and $f(x)=-1$.

A support vector is a training observation on or inside the fitted margin; it has a nonzero dual coefficient and can affect the fitted boundary.