Math 277A Rayan Saab

MATH 277A Fall 2018
Mathematical Methods in Data Science

Rayan Saab
rsaab(-at-)math.ucsd.edu
Office: AP&M 5157
Office Hours:
( by appointment)

Homework 1 Deadline: Friday October 19

Homework 2 Deadline: Tuesday December 11

A review/introduction to principal component analysis and the singular value decomposition -- Lecture notes. (May contain errors, use at your own risk).
Representation using fixed bases (e.g., Wavelets, Fourier Basis, ...) -- Lecture notes on the DFT and on Wavelets. (May contain errors, use at your own risk).
- References:
  - Cooley and Tukey's FFT paper .
  - Naoki Saito's lecture notes.
  - Blum, Hopcroft, and Kannan's Book Chapter 11.
  - Bogess's Book on wavelets.
Dictionary learning
- References:
  - Bao et al.Dictionary learning for sparse coding.
  - Aharon et al. K-SVD algorithm
Dimensionality reduction via the Johnson Lindenstrauss Lemma
- References:
  - the textbook "A mathematical introduction to compressive sensing" by Simon Foucart, Holger Rauhut (Specifically Chapter 9.5 and parts of Chapters 7 and 8 for probability background)-- and the references and notes therein.
Applications of the JL Lemma
Sparse representation and compressed sensing
- References:
  - textbook "A mathematical introduction to compressive sensing" by Simon Foucart, Holger Rauhut -- and consequently on the references and notes therein:
    - Basis Pursuit and the Null Space Property: Chapter 4
    - Mutual Coherence: Chapter 5
    - The Restricted Isometry Property and Basis Pursuit: Chapter 6
    - Gaussian and sub-Gaussian matrices: Chapter 9
    - Bounded Orthonormal Systems : Chapter 12
Nearest neighbor searches
Clustering

Textbook The book "A mathematical introduction to compressive sensing" by Simon Foucart, Holger Rauhut is very useful (but not required). It is available as an e-book via the UCSD library.
Rice University's Compressive Sensing Resources (click here)
The Nuit Blanche blog. A searchable and extensive blog on compressed sensing and related fields.

Extensions to PCA:
- Robust PCA
- Multilinear PCA (e.g., as in this paper)
- Principal Manifolds (e.g., this paper)
2D/3D Representation Systems:
- Curvelets
- Shearlets (see also this)
Dictionary Learning:
- Bao et al.Dictionary learning for sparse coding.
- Aharon et al. K-SVD algorithm