# Kernel PCA

## What's that?

• a non-linear variant of the normal linear PCA
• we intuitively map our data to a higher-dimensional feature space using some mapping f, in which it can better be linearly separated
• but we actually never specify this mapping f, but use the Kernel trick, i.e., specify some function K such that K(x,y) = <f(x),f(y)> where <.,.> denotes the inner product
• so if our algorithm does only need to compute the inner product of vectors and not f(x) explicity for some points x, we can specify a corresponding Kernel function K and avoid to specify the mapping f

## Wait... Is it about dimension reduction or dimension increasing?

• we can do dimension reduction with this technique!
• we only map the data to a higher dimensional space, to find a better projection / representation in that space using some of the principal components there
• i.e.: original space –> feature space –> some few principal components space

## What are the differences to linear PCA?

We do not compute the principal components themselves, but only the projections of our data onto those components!

## Steps of the algorithm

See here

The following slide is from this video by Pratik Prabhanjan Brahma perfectly summarize the steps for Kernel PCA: