Applying the "kernel trick" to linear methods?

Question

The kernel trick is used in several machine learning models (e.g. SVM). It was first introduced in the "Theoretical foundations of the potential function method in pattern recognition learning" paper in 1964.

The wikipedia definition says that it is

a method for using a linear classifier algorithm to solve a non-linear problem by mapping the original non-linear observations into a higher-dimensional space, where the linear classifier is subsequently used; this makes a linear classification in the new space equivalent to non-linear classification in the original space.

One example of a linear model that has been extended to non-linear problems is the kernel PCA. Can the kernel trick be applied to any linear model, or does it have certain restrictions?

Answer 1

The kernel trick can only be applied to linear models where the examples in the problem formulation appear as dot products (Support Vector Machines, PCA, etc).

Answer 2

Two further references from B. Schölkopf:

• Schölkopf, B. and Smola, A.J. (2002). Learning with kernels. The MIT Press.
• Schölkopf, B., Tsuda, K., and Vert, J.-P. (2004). Kernel methods in computational biology. The MIT Press.

and a website dedicated to kernel machines.

Answer 3

@ebony1 gives the key point (+1), I was a co-author of a paper discussing how to kernelize generalised linear models, e.g. logistic regression and Poisson regression, it is pretty straightforward.

G. C. Cawley, G. J. Janacek and N. L. C. Talbot, Generalised kernel machines, in Proceedings of the IEEE/INNS International Joint Conference on Neural Networks (IJCNN-2007), pages 1732-1737, Orlando, Florida, USA, August 12-17, 2007. (www,pdf)

I also wrote a (research quality) MATLAB toolbox (sadly no instructions), which you can find here.

Being able to model the target distribution is pretty useful in uncertainty quantification etc. so it is a useful (if fairly incremental) addition to kernel learning methods.

Answer 4

I wonder if we do not have an elefant in the room here, called kernel ridge regression, to add to the examples already cited.

1. ridge regression is a regularized version of linear regression with regularizing parameter $\lambda$ (weight decay). When $\lambda=0$ ridge regression and linear regression should be equivalent. It is a linear model.

2. kernel ridge regression is a kernelized version of ridge regression. If we put $\lambda=0$ again we have a kernelized version of linear regression (probably not very useful).

I think adding this example to the list may be helpful for the reader of this question.