The goal of the course is to present mathematical methods for the description and analysis of high dimensional data sets, with emphasis on spectral methods. We will present some of these methods, with applications to image and signal processing, structural biology, search and data mining, and more. We will explain what these methods are good for, what are their limitations, and what is the underlying math, so we will develop a good sense of when to apply them, and have a sound basis for developing new algorithms. In particular, the course will equip students with the required mathematical background to independently cope with recent developments in the field.

Topics covered in the course:

1. Introduction: curse of dimensionality, low dimensional (non-linear) structures, manifolds, Intrinsic / extrinsic dimension, dimensionality reduction
2. Linear methods: PCA, PCA in high dimensions, computational issues, linear modelling and partial least squares
3. Nonlinear methods: Laplacian eigenmaps, diffusion maps, diffusion distance, application dependent constructions.
4. Laplacians and their relation to dimensionality reduction: graph-Laplacians, estimating the dimensionality of a dataset, normalizations, convergence.
5. Spectral clustering