CS7540 Spectral Algorithms

Spring 2017, TuTh 3:05pm - 4:25pm in CoC 102

Spectral (adjective): of or relating to a specter; ghostly; phantom.


Course Information

Course Description

Spectral methods combine mathematical and algorithmic insights. This course will discuss on this connection through graphs and their spectrum, or eigenvalues. Topics of focus are random walks, clustering/graph partitioning, numerical algorithms for big data, and recent advances in graph theoretic algorithms. Presentation will assume some familiarity with graph theory (paths, cuts), probability (expectation, Markov inequality), and linear algebra (vectors, matrices, norms).


Tue Jan 10 Introduction: perfect matching and matrix rank notes notes by Pawel Parys
Thu Jan 12 Graph Laplacians and the matrix-tree theorem notes
Tue Jan 17 Eigenvalues and chromatic number notes Richard away, guest lecture by Anup B. Rao
Thu Jan 19 Random walks in Zd notes Richard away, guest lecture by Antonio Blanca
Tue Jan 24 Random walks and electrical networks notes Monograph on this subject by Doyle and Snell.
Thu Jan 26 Return probabilities of 2D and 3D grids notes Problem Set 1 Out

Tue Jan 31 Random walks and eigenvalues. notes
Thu Feb 2 Perron Frobenius Theorem (for directed graphs) Richard away, guest lecture by Anup B. Rao notes
Tue Feb 7 Cheeger's inequality and spectral clustering notes
Thu Feb 9 Expanders notes
Tue Feb 14 Eigenvalues of planar graphs notes paper circle packing
Thu Feb 16 Semi-definite programming and graph cuts notes Problem Set 2 Out

Tue Feb 21 Low rank Approximations notes Chapter 1 of Spectral Algorithms k-means clustering
Thu Feb 23 Max-CSP via Low-Rank Approximation notes Chapter 5 of Spectral Algorithms
Tue Feb 28 Approximate matrix multiplication notes Chapter 6.1 - 6.3 of Spectral Algorithms Problem Set 1 Due
Thu Mar 2 Low Rank Approximations of Tensors notes Chapter 8.1 of Spectral Algorithms
Tue Mar 7 Random projections notes
Thu Mar 9 Subspace embeddings notes booklet on this topic Problem Set 2 Updated
Tue Mar 14 Matrix concentration bounds notes matrix concentration survey OSE via Matrix Concentration
Thu Mar 16 Graph sparsification Problem Set 3 Out notes
Tue Mar 21 No class due to Spring Break
Thu Mar 23 No class due to Spring Break

Tue Mar 28 Iterative methods for solving linear systems notes
Thu Mar 30 Linear system solvers via sparsification Richard away, lecture by Peng Zhang. Dan Spielman's notes notes on solvers
Tue Apr 4 More sparsification based algorithms notes on solvers
Thu Apr 6 Low stretch spanning trees and subgraph preconditioners notes on solvers
Tue Apr 11 Gradient descent
Thu Apr 13 Non-linear preconditioning and maximum flows Problem Set 2 Due notes
Tue Apr 18 Oblivious routings notes

Thu Apr 20 Generalized preconditioning notes
Tue Apr 25 reserved for presentations Last day of classes
Thu May 4 Problem Set 3 Due

Some Ideas for Projects

Many of the algorithmic results covered in this course make heavy use of randomization. Derandomized version of these with better parameters have a wide range of important applications. Some keywords that may be good start points for Googling are: (links to some representative papers will be added soon)