Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

Module MAT-65-15-M-7

Module Identification

Module Number Module Name CP (Effort)
MAT-65-15-M-7 Optimization on Manifolds 9.0 CP (270 h)

Basedata

CP, Effort 9.0 CP = 270 h 2 Sem. from irreg. [7] Master (Advanced) [EN] English Steidl, Gabriele, Prof. Dr. (PROF | DEPT: MAT) Steidl, Gabriele, Prof. Dr. (PROF | DEPT: MAT) [MAT-SPAS] Analysis and Stochastics [MAT-88.105-SG] M.Sc. Mathematics [AUSL] Phase-out period

Notice

The module can be divided into the modules [MAT-65-15A-M-7] "Optimization on Manifolds - Part 1" and „Optimization on Manifolds - Part 2“ (each 4.5 Cp).

Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
2V+1U MAT-65-15A-K-7
Optimization on Manifolds - Part 1
P 42 h 93 h - - PL1 4.5 irreg.
2V MAT-65-15B-K-7
Optimization on Manifolds - Part 2
P 28 h 107 h - - PL1 4.5 irreg.
• About [MAT-65-15A-K-7]: Title: "Optimization on Manifolds - Part 1"; Presence-Time: 42 h; Self-Study: 93 h
• About [MAT-65-15B-K-7]: Title: "Optimization on Manifolds - Part 2"; Presence-Time: 28 h; Self-Study: 107 h

Examination achievement PL1

• Form of examination: oral examination (20-30 Min.)
• Examination Frequency: irregular (by arrangement)
• Examination number: 86351 ("Optimization on Manifolds")

Evaluation of grades

The grade of the module examination is also the module grade.

Contents

• Gateaux and Fréchet differentiability,
• differentiable finite-dimensional manifolds: charts, atlas, tangent vectors and tangent spaces, vector fields, differential of a mapping,
• Riemannian metric, geodesics, Christoffel symbols, exponential map and logarithmic map,
• Hopf-Rinow theorem,
• first-order optimization, descent methods,
• examples of manifolds: spheres, hyperbolic spaces, positive definite matrices, probability simplex, Grassmann manifolds, Stiefel manifolds, special Euclidean group, rotation group,
• linear connection and parallel transport.
• Levi-Civita connection,
• curvature,
• Hessian,
• second-order optimization methods, Newton-like methods,
• Jacobi fields,
• Hadamard manifolds: introduction, proximal point methods, optimization methods from convex analysis,

• (possible outlooks: Lie groups and Lie algebras, integration on manifolds).

Competencies / intended learning achievements

Upon successful completion of this module, the students have studied and understand fundamental methods and propositions of the theory of Riemannian manifolds. After attending the first part of the course, they are familiar with first-order optimization and descent methods on manifolds. They have learnt important examples of manifolds and gained an impression of the role of manifolds in the context of image processing and data analysis. After finishing the second part of the course, the students have broadened their knowledge and know more effective second-order optimization methods on manifolds. In addition, they have learnt how methods of convex analysis can be generalized to Hadamard manifolds.

They understand the mathematical background of the algorithms and are able to analyze them and to apply them to concrete questions. They understand the proofs presented in the lecture and are able to comprehend and explain them. In particular, they are able to outline the conditions and assumptions that are necessary for the validity of the statements. By completing exercises, the students have developed a skilled, precise and independent handling of the terms, propositions and techniques taught in the lecture. Moreover, they have performed implementations for the application of the algorithms in image processing.

Literature

• P.-A. Absil, R. Mahony, R. Sepulchre: Optimization Algorithms on Matrix Manifolds,
• J. Jost: Riemannian Geometry and Geometric Analysis,
• J. M. Lee: Introduction to Smooth Manifolds,
• S. Helgason: Differential Geometry, Lie Groups and Symmetric Spaces,
• D. Gromoll, W. Klingenberg, and W. Meyer: Riemannsche Geometrie im Großen.

Materials

Further literature will be announced in the lecture.

None

References to Module / Module Number [MAT-65-15-M-7]

Module-Pool Name
[MAT-65-MPOOL-7] Specialisation Image Processing and Data Analysis (M.Sc.)
[MAT-AM-MPOOL-7] Applied Mathematics (Advanced Modules M.Sc.)