Example
Consider three concentric clouds of points (shown); we wish to use kernel PCA to identify these groups. The color of the points is not part of the algorithm, but only there to show how the data groups together before and after the transformation.
First, consider the kernel
Applying this to kernel PCA yields the next image.
Now consider a Gaussian kernel:
That is, this kernel is a measure of closeness, equal to 1 when the points coincide and equal to 0 at infinity.
Note in particular that the first principal component is enough to distinguish the three different groups, which is impossible using only linear PCA, because linear PCA operates only in the given (in this case two-dimensional) space, in which these concentric point clouds are not linearly separable.
Read more about this topic: Kernel Principal Component Analysis
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