Spectral Analysis of Data

In this lecture, we discuss some spectral techniques and their applications. The goal of spectral techniques is to study and deduce the characteristics of a matrix by looking at its spectrum (i.e. eigenvalues of matrix). Although spectral methods can be applied to any problem instance with matrix representation, we will be focussing on graph theoretic problems in this lecture. Over the past two decades, spectral techniques have proven to be quite useful in analysing some of the graph problems like coloring, bisection, max-cut. The aim of this lecture is to give the reader an intuition why spectral techniques are giving such good results and discuss a couple of applications. We would also like to mention that recently Prof. Daniel Spielman (Yale University) gave a tutorial on spectral graph theory in FOCS 2007 conference. His slides, presentation data and an online recording of his lecture are available at [7] and can be viewed freely. We encourage you to go through them. These notes are organized as follows. In the next section, we discuss an application of spectral analysis on page ranking. In section 3, we discuss some intuitions about spectral techniques and how they are applied to some applications. This is followed by a discussion on application of spectral methods in graph coloring in section 4. We conclude in section 5.