Lecture Notes

Lecture notes are courtesy of MIT students and are used with permission.

Spectral Graph Theory
1 Linear algebra review, adjacency and Laplacian matrices associated with a graph, example Laplacians ( PDF )
2 Properties of the Laplacian, positive semidefinite matricies, spectra of common graphs, connection to the continuous Laplacian ( PDF )
3 Courant-Fischer and Rayleigh quotients, graph cutting, Cheerger’s Inequality ( PDF )
4 (Lazy) random walks, their stationary distribution and l2-convergence, normalized Laplacian, conductance, Monte Carlo methods ( PDF )
5 Monte Carlo methods continued, approximate DNF counting, approximating the permanent of 0-1 matrices ( PDF )
6 Diameters and eigenvalues, expander graphs ( PDF )
7 Nonblocking routing networks, local and almost-linear time clustering and partitioning, Lovasz-Simonovits Theorem ( PDF )
8 Local and almost-linear time clustering and partitioning (cont.), PageRank, introduction to sparsification ( PDF )
9 Sparsification (combinatorial and spectral), effective resistance, matrix pseudoinverses and tail bounds ( PDF )
10 Spectral sparsification (cont.), introduction to convex geometry ( PDF )
Convex Geometry
11 Polar of a convex body, separating hyperplanes, norms and convex bodies, Banach-Mazur distance, Fritz John’s theorem ( PDF )
12 Separating hyperplanes (cont.), Banach-Mazur distance, Fritz John’s theorem, Brunn-Minkowski inequality ( PDF )
13 Brunn-Minkowski inequality (cont.), Brunn’s theorem, isoperimetric inequality, Grunbaum’s theorem ( PDF )
14 Approximating the volume of a convex body ( PDF )
15 Random sampling from a convex body (cont.), grid walk, introduction to concentration of measure ( PDF )
16 Concentration of measure and the isoperimetric inequality, Johnson-Lindenstrauss theorem ( PDF )
17 Johnson-Lindenstrauss theorem (cont.), Dvoretsky’s theorem ( PDF )
Lattices and Basis Reduction
18 Lattices, fundamental parallelepiped and dual of a lattice, shortest vectors, Blichfield’s theorem ( PDF )
19 Minkowski’s theorem, shortest/closest vector problem, lattice basis reduction, Gauss’ algorithm ( PDF )
20 LLL algorithm for lattice basis reduction, application to integer programming ( PDF )
Iterative Methods for Linear Algebra
21 Iterative methods to solve linear systems, steepest descent ( PDF )
22 Convergence analysis of steepest descent and conjugate gradients ( PDF )
23 Preconditioning on Laplacians, ultra-sparsifiers ( PDF )
Multiplicative Weights
24 Multiplicative weights ( PDF )
25 Multiplicative weights and applications to zero-sum games, linear programming, boosting, and approximation algorithms ( PDF )

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