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 |
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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 Z_{d} | 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 |