Just like matrix Chernoff bounds were a generalization of scalar Chernoff bounds, the multiplicative weights algorithm can be generalized to matrices. Recall that in the setup for the multiplicative weight update algorithm, we had a sequence of time steps t = 1, . . . , T ; in each time step t, we made a decision i ∈ {1...N} and got a value vt(i) ∈ [0, 1]. After we made a decision in time step ...

In this third talk we will discuss properties related to edge expansion. In particular, we will define the Cheeger constant (which measures how easy it is to cut off a large piece of the graph) and state the Cheeger inequalities. We also will define and discuss discrepancy for undirected and directed graphs. We also state the Perron-Frobenius Theorem which is a useful tool in spectral graph the...

In this first talk we will introduce three of the most commonly used types of matrices in spectral graph theory. They are the adjacency matrix, the combinatorial Laplacian, and the normalized Laplacian. We also will give some simple examples of how the spectrum can be used for each of these types.

As one approach to uncovering the genetic underpinnings of complex disease, individuals are measured at a large number of genetic variants (usually SNPs) across the genome and these SNP genotypes are assessed for association with disease status. We propose a new statistical method called Spectral-GEM for the analysis of genome-wide association studies; the goal of Spectral-GEM is to quantify th...

We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator T...

Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. Laplace’s equation and its discrete form, the Laplacian matrix, appear ubiquitously in mathematical physics. Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial optimization, computer vision, computer graphics, and machine le...

We also saw that λ2 = minR(y). The issue is that we may have vol(St) > vol(V −St). To fix this, we will modify y so that vol(supp(y)) ≤ m (recall that vol(V ) = 2m). The idea is to pick c such that the two sets {i : y(i) < c} and {i : y(i) > c} both have volume at most m, then find St for both of them and take the best one. This lecture is derived from Lau’s 2012 notes, Week 2, http://appsrv.cs...

We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of P2 and Pn−2 is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetković and Rowlinson from 1...

Graph partitioning problems are NP-complete and various heuristic algorithms exist in the litterature. Particularly, spectral graph partitioning algorithms partition the graph using the eigenvector associated with the second smallest eigenvalue of the \graph Laplacian." Through the use of graph theory we have devoloped preconditioned subspace algorithms for spectral partitioning.

Our construction of error-correcting codes will exploit bipartite expander graphs (as these give a much cleaner construction than the general case). Let’s begin by examining what a bipartite expander graph should look like. It’s vertex set will have two parts, U and V , each having n vertices. Every vertex will have degree d, and every edge will go from a vertex in U to a vertex in V . In the s...