# Coursebooks 2018-2019

## Low-rank approximation techniques

Kressner Daniel

English

#### Summary

Low-rank approximation techniques have become a key tool in scientific computing to deal with large-scale problems and high-dimensional data. This course covers state-of-the-art algorithms and current research in this area.

#### Content

- Theoretical background of low-rank matrix approximation
- Subspace iteration
- Randomized low-rank approximation
- Low-rank approximation by deterministic column/row selection
- Low-rank approximation by randomized sampling
- Basic introduction to tensors
- Tensor rank, CP, Tucker, and TT decompositions of tensors
- Alternating least-squares algorithms
- Riemannian optimization on low-rank matrix and tensor manifolds

#### Keywords

numerical algorithms, linear algebra, matrix, tensor, random vectors, high dimensions, low rank

#### Learning Prerequisites

##### Required courses

Linear Algebra, Numerical Analysis

##### Recommended courses

Probability theory

##### Important concepts to start the course

Programming in Matlab, Python, Julia, or a similar language.

#### Learning Outcomes

By the end of the course, the student must be able to:
• Choose a suitable low-rank approximation techniques for treating a large-scale problem or high-dimensional data
• Analyze algorithms for low-rank approximation
• Prove fundamental results in low-rank approximation
• Implement low-rank approximation algorithms

#### Transversal skills

• Plan and carry out activities in a way which makes optimal use of available time and other resources.
• Use a work methodology appropriate to the task.
• Assess one's own level of skill acquisition, and plan their on-going learning goals.
• Demonstrate a capacity for creativity.
• Write a scientific or technical report.

#### Teaching methods

Lectures and exercises.

#### Expected student activities

Attending lectures, exercises, and doing a mini-project.

#### Assessment methods

Oral exam covering key concepts of the course. During the oral exam, the mini-project, which accounts for 20% of the grade, will be evaluated.

#### Supervision

 Office hours No Assistants Yes Forum No

#### Resources

##### Bibliography

References to the current literature will be provided in the slides and lecture notes. Many of the linear algebra foundations of this course are contained in Horn/Johnson: Matrix Analysis, 2nd edition, CUP, 2012.

##### Notes/Handbook

Detailed slides and lecture notes will be provided as the course progresses.

### Reference week

MoTuWeThFr
8-9 MAA110
9-10
10-11 MAA110
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22

Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German