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Kernel principal component analysis diagonalizes the centered $n$ by $n$ kernel matrix instead of an explicit covariance operator in a possibly infinite-dimensional feature space. A new point has coordinateso the leading coordinates require only evaluations of the positive-semidefinite kernel.

Running the K-nearest neighbors algorithm in a truncated collection of these coordinates can remove low-variance noise, reduce effective dimension, and allow a nonlinear boundary in the original covariates. This is the kernel trick: every feature-space inner product needed for fitting and projection is replaced by $k(x,z)=⟨\varphi (x),\varphi (z){⟩}_{\mathcal{H}}$ without constructing $\varphi (x)$ explicitly.

Kernel principal component analysis can represent nonlinear low-dimensional structure by performing linear PCA in a nonlinear feature space. It can also work directly with structured objects such as strings through a kernel, without assigning them explicit finite-dimensional coordinates. Both capabilities are unavailable to ordinary linear PCA on the original variables.