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Unit M15: Applied Numerical Analysis

John von Neumann Professor Mauro Maggioni

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John von Neumann Lecture

Statistical Learning, Geometry and High-Dimensional Probability (Statistische Lerntheorie, Geometrie und hochdimensionale Wahrscheinlichkeitstheorie)

We will cover foundations of random matrix theory (with applications to compressed sensing and signal processing), nonparametric statistical estimation and machine learning, and problems about the geometry of high dimensional data sets.
Random matrices. Basic theory of random matrices: basic concentration inequalities, subgaussian random variables, singular values of random matrices. Applications: to compressed sensing theory; to numerical linear algebra (a.k.a. how to compute quickly highly accurate low-rank approximate Singular Value Decompositions, with high probability).
Nonparametric estimation: Basic results in nonparametric density estimation and nonparametric regression in low dimensions. Obstructions in the high-dimensional setting, curse of dimensionality. Applications: denoising of signals.
Approximation theory. A primer in nonlinear approximation of functions, especially for wavelets and other multiscale approximations. Multiscale approximation of functions in high dimensions. Attacking the curse of dimensionality.
Multiscale Analysis in High dimensions. Multiscale geometric constructions in metric spaces, associated algorithms and applications. Multiscale SVD and Geometric Multiresolution analyses, and their applications to dictionary learning, regression, manifold learning, compressive sensing.
Optimal transport. A primer in optimal transport theory and Wasserstein metrics between distributions. Current research: multiscale approximation theory in the space of probability measures with respect to Wasserstein metrics.

Educational objectives:
After successful completion of the module, the students will master key tools in probability applied to high- dimensional settings, including basic random matrix theory, approximation of functions in high-dimensions, and other problems in statistical learning theory and high-dimensional data analysis.

Educational methods: lectures, assigned exercises, reading of research papers
Media: blackboard lectures, slides

Level: Master
Duration: 1 Semester (SS 14), lectures will be held from May to July on 17 dates with weekly exercises
Language: English
Exam: written (90 min.)

Prerequisites (recommended): Linear Algebra; Real Analysis; Probability. Helpful but not required: functional analysis (especially Hilbert spaces); numerical linear algebra; stochastic calculus.

Literature: It will be in the form of online lecture notes, papers, books. It will be provided at the beginning of class.

-- SteffenPeter - 12 Mar 2014