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A kernel support vector machine constructs a large-margin classifier in a feature Hilbert space. Given training points $x_i$ and labels $y_i\in \{-1,1\}$, choose a feature map $\Phi :X\to \mathcal{H}$ and an affine score $f(x)=⟨w,\Phi (x)⟩+b$. Prediction is the sign of the score. Normalizing the functional margin to one makes the closest separating geometric margin $1/\Vert w\Vert$, and the full margin between the two supporting hyperplanes is $2/\Vert w\Vert$. Maximizing that margin therefore minimizes $\frac{1}{2}\Vert w{\Vert }^2$.

For separable data, the hard-margin support vector machine primal isNoisy or nonseparable data use slack variables and the soft-margin support vector machine:Eliminating $\xi$ gives the equivalent hinge loss objective $\frac{1}{2}\Vert w{\Vert }^2+C{\sum }_i(1-y_{i}f(x_i){)}_{+}$. The parameter balances margin size against violations; it is not a hard bound on their number. The bias is unpenalized here, which determines the equality constraint in the dual.

Attach multipliers ${\alpha }_i\ge 0$ to $1-{\xi }_i-y_{i}f(x_i)\le 0$ and ${\eta }_i\ge 0$ to $-{\xi }_i\le 0$. The Lagrangian isStationarity gives $w={\sum }_i{\alpha }_i{y}_i\Phi (x_i)$, ${\sum }_i{\alpha }_i{y}_i=0$ and $C-{\alpha }_i-{\eta }_i=0$. Eliminating $w,b,\xi$ yieldswhere $k(x,x^{\prime })=⟨\Phi (x),\Phi (x^{\prime })⟩$ is a positive-definite kernel, meaning positive semidefinite Gram matrices. The hard-margin dual is the same objective with only ${\alpha }_i\ge 0$, without the upper bound. Soft-margin strict feasibility follows by taking $w=b=0$ and all ${\xi }_i>1$, so strong duality and the Karush-Kuhn-Tucker conditions apply. For separable hard-margin data a separator can be rescaled to give strict margins, supplying the analogous qualification. Although the feature space may be infinite dimensional, projecting $w$ onto the finite span of the training feature vectors preserves scores and cannot increase its norm, so this optimization reduces to a finite span.

The complementary slackness relations arePoints with ${\alpha }_i>0$ are support vectors. If $0<{\alpha }_i<C$, then ${\xi }_i=0$ and $y_{i}f(x_i)=1$; such a point gives $b=y_i-{\sum }_j{\alpha }_j{y}_{j}k(x_j,x_i)$. A point with ${\alpha }_i=C$ can lie inside the margin or be misclassified, while ${\alpha }_i=0$ contributes no term to $w$. If no coefficient lies strictly between the bounds, an admissible bias must be obtained from the KKT inequalities, rather than dividing by a nonexistent margin vector. Bias and dual coefficients can be nonunique even when the optimal feature-space weight is unique.

The kernel trick operates both during training and prediction. The dual optimization uses only the Gram matrix $K_{ij}=k(x_i,x_j)$, so the feature vectors need not be formed. Prediction likewise usesA nonlinear kernel therefore makes a linear separator in feature space represent a nonlinear boundary in input space. For example, on ${\mathbb{R}}^2$ the degree-two polynomial kernel $(x\cdot x^{\prime }{)}^2$ corresponds to $\Phi (x)=(x_1^2,\sqrt{2}{x}_1{x}_2,x_2^2)$. Gaussian kernels yield infinite-dimensional feature spaces. An arbitrary similarity is not automatically a valid kernel: for every finite collection and real coefficients $a_i$, one needs ${\sum }_{i,j}{a}_i{a}_{j}k(x_i,x_j)\ge 0$. This makes the dual quadratic form positive semidefinite and the maximization concave.

Mercer's theorem gives a spectral realization under its additional analytic hypotheses. For a continuous symmetric positive-semidefinite kernel on a compact domain with a finite full-support measure, the integral operator $(T_{k}f)(x)=\int k(x,x^{\prime })f(x^{\prime })d\mu (x^{\prime })$ is compact, self-adjoint and positive. The Mercer expansion iswith the standard uniform convergence conclusions under these hypotheses. Since ${\sum }_j{\lambda }_j{e}_j(x{)}^2=k(x,x)<\infty$, the nonlinear feature mapis well defined. This explains the connection of Mercer kernels to Hilbert space features, rather than treating the kernel trick as a purely formal substitution.

The finite-Gram positivity condition is more general than this compact-domain spectral theorem. Every such kernel generates a Reproducing-kernel Hilbert space: on finite sums of kernel sections definequotient out zero-norm elements, and complete. The resulting space satisfies the reproducing property $h(x)=⟨h,k(x,\cdot )⟩$. Its canonical feature map is $\Phi (x)=k(x,\cdot )$, whose inner product is exactly $k(x,x^{\prime })$. A feature map need not be injective; calling it an embedding does not by itself prove distinct inputs remain distinct. The kernel support vector machine uses this geometry together with convex duality to fit and evaluate a maximum-margin classifier while accessing the geometry only through kernel evaluations.