# Low-rank tensor approximation

**Lecturer:**André Uschmajew**Date:**Thursday 11:00 - 12:30**Room:**MPI MiS G3 10**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Prerequisites:**Basic knowledge about linear algebra, analysis and functional analysis

## Abstract:

The concept of matrix rank is fundamental in linear algebra, and the idea of low-rank matrix approximation has found many useful applications in modern applied mathematics. For instance it is used for data analysis and compression. The singular value decomposition is the central tool for this task and the starting point for a rich theory of low-rank matrices that combines linear algebra with analysis and geometry.

If one aims at a similar theory for higher-order tensors, one is faced with several options to generalize the notion of rank, and for any of these the low-rank approximation task is considerably harder. In this lecture we consider some of these generalizations, including low-rank tree tensor networks like the (hierarchical) Tucker format and the tensor train format. For them, the concept of higher-order singular value decomposition plays a similar role for low-rank approximation like the SVD in the matrix case. It is a main goal of this lecture to get familiar with this concept.

Additionally, some special topics of interest will be discussed, e.g., manifold structure of low-rank matrices, spectral norm of tensors, tensor products of operators, and motivating low-rank approximation tasks arising in high-dimensional scientific computing. Lectures will be as self-conatined as possible.

## Regular Lectures (Summer 2018)

**Geometric and topological methods of data analysis**- Jürgen Jost
- Friday, 13:30 - 15:00, MPI MiS A3 01

**Regularity theory for nonlinear, degenerate partial differential equations**- Benjamin Gess
- Monday, 16:15 - 17:45, MPI MiS A3 01

**The signature of iterated integrals: algebra, analysis and machine learning**- Joscha Diehl
- Thursday, 09.00-10.30, A3 02

**Geometric Variational Problems**- Ulrich Menne
- Monday & Friday, 9:15 - 10:45, Augusteum, Room 314

**The Calderón Problem**- Angkana Rüland
- Thursday, 13:15-14:45, Augusteum, Room 314

**Introduction to the KPZ Equation**- Scott Smith
- Monday, 10:30 - 12:30, MPI MiS A3 01

**Low-rank tensor approximation**- André Uschmajew
- Thursday, 11:00 - 12:30, MPI MiS G3 10

**Data: crush course**- Ivan P. Yamshchikov
- Friday, 12.00-13.30, A3 02

**Stochastic homogenization**- Felix Otto
- Wednesday, 09:15 - 11:00, MPI MiS A3 01

**Reading seminar on “J-holomorphic curves”**- Enno Keßler and Ruijun Wu
- Wednesday, 13:00 - 14:30, MPI MiS A3 01

**IMPRS-Ringvorlesung: Introduction to Nonlinear Algebra**- Bernd Sturmfels and Mateusz Michałek
- Tuesday, 10:00 - 12:00, MPI MiS G3 10