What is the kernel trick in support vector machines?

• A. Maps data into higher-dimensional space explicitly to separate with a linear hyperplane.
• B. Maps data into higher-dimensional space explicitly to separate with a non-linear boundary.
• C. Maps data into higher-dimensional space implicitly to separate with a linear hyperplane; example: RBF kernel k(x,y)=exp(-γ||x-y||^2).
• D. It is never used with SVM.

Answer: (C)

The kernel trick lets you work in a high-dimensional feature space without explicitly mapping data there, so the separation can be linear in that space while appearing nonlinear in the original input space. You compute inner products in that high-dimensional space with a kernel function, k(x, y) = ⟨φ(x), φ(y)⟩, which lets the SVM learn a linear decision boundary in the feature space without ever constructing φ(x). A classic example is the RBF kernel, k(x, y) = exp(-γ||x − y||^2), which corresponds to mapping into an infinite-dimensional space while only requiring kernel evaluations. This is why the kernel trick is powerful: it makes complex, nonlinear boundaries in the input space tractable by exploiting linear separation in a cleverly chosen feature space.