| 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) |
