TR 9:30-11am, Room: Klaus 1447

Instructor: Santosh Vempala

Spectral methods have transcended their origins in linear algebra and
numerical analysis and are now used in almost every domain with large data sets from
web search (e.g., pagerank) and information retrieval (e.g., LSI) to medical
diagnosis and prediction. This course is about the mathematics and the
algorithmic insights driving these applications.

**Tentative schedule**

Notes (in progress; updated frequently)

Jan 8. Intro/overview.

Jan 10. SVD basics, a geometric characterization; P1: k-means.

Jan 15. P2: Clustering random models; planted clique/partitions

Jan 17. P2 (contd.). Clustering under deterministic assumptions. K-means revisited.

Jan 22, 24. P3: Unraveling Mixture Models via PCA

Jan 29, 31. Learning Mixture models and topic models via higher moments

Feb 5, 7. P4: Independent Component Analysis using moments

Feb 12, 14. Algo 1: Matrix approximation by random sampling.

Feb 19,21. **Guest Lectures by Ravi Kannan**

A2: Cut decompositions, Weak regularity; P5: Constraint Satisfaction Problems
A3: Tensor decompositions via sampling

Feb 26,28. A4: Combinatorial Optimization via Low-rank Approximation.

Mar 5, 7. P6: Cheeger cuts; P7: Clustering with worst-case guarantees

Mar 12, 14. Mar 7. A5: Constant-time approximate SVD
Adaptive sampling A6: Volume sampling, A7: Isotropic Random Projection

Mar 19, 21. Spring break

Mar 26, 28. A8: CUR decomposition, P8: Recommendation systems.

Apr 2, 4. P3 (revisited): Mixture models, A9: Affine-invariant PCA.

Apr 9,11. A10: Noise-tolerant PCA

Apr 16, 18. A11: Sparsification via length-squared sampling. P9: Solving linear equations.

Apr 23, 25. P10: Learning convex sets, TBD.