Topics for projects - Ronny Bergmann

In my research I am interested in differential geometry and Optimization. Given a Riemannian manifold \(\mathcal M\), for example the sphere, we are given the task to find a minimizer of a function \(f\colon \mathcal M \to \mathbb R\)

Informally, working on a Riemannian manifold, means that “locally” the space we are in still looks like some \(\mathbb R^n\), but globally it does not. We, however, still have a certain structure on this space, that allows us to consider “looking directions” (tangent vectors) and measure angles and lengths of these.

Examples are

• the already mentioned Sphere \(\mathbb S^n\) of unit vectors in \(\mathbb R^{n+1}\)
• the manifold of rotation matrices \(\mathrm{SO}(n)\)
• the Stiefel manifold \(\mathrm{St}(n,p)\) of \(p\) orthonormal vectors in \(\mathbb R^n\)
• the symmetric positive definite (spd) matrices \(\mathcal P(n)\)

as a concrete example, in Diffusion tensor MRI (DTMRI) every measurement in every point of a 3D measurement space is a spd matrix. We obtain a 2D grid or an image of data, where every “pixel” is an spd matrix. This actually is the (power) manifold \(\mathcal M = \mathcal P(3)^{m\times n}\), where \(m,n\) a the width and height of the image in pixels, respectively.

When now consider optimization tasks, maybe just a simple gradient descent, two major points to discuss are, that first of all two points $p, q \in \mathcal M$ on a manifold can not just be added, for example adding two unit vectors is in general not a unit vector, but that second, the definition of the classical gradient heavily depends on this possibility to add elements.

Topics I offer are for example from the areas

• certain manifolds and numerical algorithms thereon, for example retractions, vector transport,…
• numerical algorithms for optimisation, for example when the cost $f$ is non-smooth, has further constraints or is a (large) sum of single functions to minimise, or discussing generalisations of Newton's method
• applications where this is used, for example the already mentioned DT-MRI, it appears in density function theory, together with Lie groups in robotics, or with alignment problems or covariance matrices in statistics
• or possibly the relation of these topics to ODEs on Lie groups.

You can see several previous topics for example in my list of supervised master theses or the list of supervised specialisation projects.

Most of these topics can include programming, where my personal favorite is Julia, using Manopt.jl and Manifolds.jl or more recently LieGroups.jl

You should have had the course TMA4180 Optimization I. Some knowledge on Differential Geometry and Riemannian manifolds would be beneficial but is not necessary. We could for example do the Specialisation Course as a reading course towards Differential Geometry and/or Optimisation alongside a Specialisation project.

If you are interested in any of the topics or would like to discuss more precise ideas of topics, please send me an email. We can also discuss ideas that you have, or preferences towards implementation or theory, and taylor a topic towards that