This seminar introduces a set of concepts, methods and tools from differential calculus in several variables with a view to optimization on differentiable manifolds for use in engineering applications. Based on examples from real applications, students in Engineering will become acquainted with rigorous mathematical techniques to understand and apply advanced differential geometric techniques in data processing, computer vision and signal processing.

With this lecture, students are expected to be familiar with the following topics:

1. Geometry of the sphere – Tangent space, geodesics, parallel transport
2. Smooth real-valued function on the sphere - Riemannian gradient, critical point, Riemannian Hessian
3. Newton-type method on the unit sphere - Formulation, convergence properties
4. Jacobi method - Eigenvalue decomposition of symmetric matrices, singular value decomposition
5. Data whitening - Principal component analysis
6. Multi-way array analysis - Generalized low rank approximation, tensor singular value decomposition
7. FastICA as Newton-type method on the sphere - Derivation, refinement, convergence properties
8. JADE algorithm - Diagonalization of cumulant tensors
9. Joint diagonalization of matrices - BSS as time-delayed covariance matrices

References:

1. P.-A. Absil, R. Mahony, R. Sepulchre: Optimization Algorithms on Matrix Manifolds. Princeton University Press, 2008.
2. U. Helmke and J.B. Moore: Optimization and Dynamical Systems, Springer-Verlag, 1993.

Lecture materials:

week 01: reading.

The seminar is open to all students who have completed their bachelor degree. Our aim is that participants learn to process scientific information and present existing results appropriately.