Lecture : Spectral Methods for Partitioning Graphs ( 1 of 2 )

Today and next time, we will cover what is known as spectral graph partitioning, and in particular we will discuss and prove Cheeger’s Inequality. This result is central to all of spectral graph theory as well as a wide range of other related spectral graph methods. (For example, the isoperimetric “capacity control” that it provides underlies a lot of classification, etc. methods in machine learning that are not explicitly formulated as partitioning problem.) Cheeger’s Inequality relates the quality of the cluster found with spectral graph partitioning to the best possible (but intractable to compute) cluster, formulated in terms of the combinatorial objectives of expansion/conductance. Before describing it, we will cover a few things to relate what we have done in the last few classes with how similar results are sometimes presented elsewhere.