Kernel support vector machine

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.