Edited by author 09-25-2001 08:35 PM
Hi all.

This paper talks about the kernel trick. The kernel trick is used in order to minimize the computational burden associated with computing dot products. The kernel trick proceeds by mapping data from input space into a higher dimensional feature space and then performing the dot product there. The problem is that once you have mapped the data into a higher dimensional space and then have separated the data, how do you return to input space? This paper specifically addresses this problem. It talks about how it does this, which is by using fixed-point iteration. This paper is important because it addresses this fundamental problem, and also because it demonstrated how it can be used for a particular application: de-noising. De-noising experiments are shown at the end.

The paper is divided into 5 sections. Section one defines what kernels are, and introduces the kernel trick. The second section talks about peforming the mapping of data from input space to feature space. Mercer theory is the mathematics of describing the conditions under which the kernel trick can be applied. The third section reviews two algorithms that are used for linear separation of data in feature space, namely kernel principal components analysis (KPCA) and support vector machines (SVMs). The fourth section is the most important, since it describes how to return from feature space to input space. The method that is used to do that is called fixed-point iteration, as mentioned above. Section 5 discusses reduced set methods. I am not going to talk about section 5 in the talk. The final section, section 6, demonstrates various experiments that are used for de-noising.

The most important point of the paper is the notion of the kernel trick, and also the method they use to return from feature space to input space. So the most important sections are 1 and 4.

My talk will be very conceptual and intuitive. I will not delve into many of the mathematical details. I will just try to provide a high-level overview of the important concepts that are covered in the paper.

Joe