# M.Sc. Naomi Schneider

## M.Sc. Naomi Schneider

office: EN-B 201 address: M.Sc. Naomi Schneider Geomathematics Group Emmy-Noether-Campus University of Siegen Walter-Flex-Str. 3 57068 Siegen | Tel: from abroad: +49 271 740 3589 from Germany: 0271 740 3589 Fax: from abroad: +49 271 74013589 from Germany: 0271 74013589 |

E-mail:

Naomi.Schneider (a) uni-siegen.de (please replace ' (a) ' by '@').

**Project description:**

My PhD-thesis deals with combining matching pursuits for inverse problems with dictionary learning techniques. This combination is realized by the so-called Learning Inverse Problem Matching Pursuit (LIPMP) algorithm. The algorithm is currently tested against EGM2008 and GRACE data for the downward continuation of the gravitational potential. This is an important problem in the geosciences as a model of this potential in high-resolution is needed, for instance, to visualize the climate change.

In the last decade, the Inverse Problem Matching Pursuit (IPMP, i.e. the Regularized Functional Matching Pursuit (RFMP) and the Regularized Orthogonal Functional Matching Pursuit (ROFMP)) have been developed in the Geomathematics Group Siegen in order to solve ill-posed inverse problems in the planetary sciences as well as medical imaging. These algorithms have been proven to construct competitive alternative methods for, e.g., the downward continuation of the gravitational potential. As matching pursuits, they iteratively construct an approximation of the gravitational potential from an a-priori chosen 'dictionary' by minimizing a Tikhonov functional. Such a dictionary can generally be considered as an overcomplete set of trial functions, such as spherical harmonics, radial basis functions and wavelets as well as scalar Slepian functions.

However, experiments show a dependence of the results on the manually chosen dictionary. Thus, the need to develop an automatized process of determining (i.e. learning) problem-specific, suitable trial functions occurs. Therefore, we want to learn a dictionary for a particular task. Moreover, we demand to learn established, physically interpretable trial functions instead of constructing new ones.

We have developed such a learning algorithm, the LIPMP. Our learning approach is a generalization of the established IPMPs to an infinite dictionary. From this infinite dictionary, a finite set of optimized candidates are computed, mostly via non-linear constrained optimization techniques. A minimizer of the Tikhonov functional among these candidates defines then a learnt dictionary element. Dependent on the given problem, several additional features have been developed to guide the learning process. In this way, the LIPMP is the first algorithm to automatically select a best basis for approximating a signal. Furthermore, this approach can be reformulated as a doubled minimization problem which is a common approach in the field of dictionary learning.

The described routine constitutes a first learning approach. In the sense of machine learning, this approach could be classified as a very simple reinforcement learning algorithm and, thus, a supervised learning ansatz. This point of view could also lead to an advanced learning algorithm in future works.

We have published first results (V. Michel, N. Schneider: *A first approach to learning a best basis for gravity field modelling*, submitted to GEM: International Journal on Geomathematics, arXiv: 1901.04222v2). They show that the RFMP terminates with smaller errors when using a much smaller, but learnt dictionary. Furthermore, a first runtime analysis promises a competitive runtime of the LIPMP.

In the future, we want to consider the following theoretical aspects:

- We solve non-linear constrained optimization problems. What do we know about them?
- The LIPMP solves an ill-posed inverse problem. What can we say about the obtained approximation?
- We seek a 'good' dictionary. How can we formulate these term mathematically?
- We obtain a dictionary from the LIPMP. Is this a good dictionary?

**I am interested in**

inverse problems, constructive approximation, optimization,

machine and dictionary learning, reinforcement learning,

geosciences and medical imaging

**Previous presentations, title**

- 09.2016: Workshop "Approximation Methods and Data Analysis" in Hasenwinkel, Vectorial Slepian Functions on the Ball
- 09.2017: Workshop "Geomathematics Meets Medical Imaging" in Speyer, Vectorial Slepian Functions on the Ball
- 04.2018: Conference "European Geosciences Union General Assembly 2018" in Vienna, Learning an optimized dictionary for gravity field modelling
- 05.2018: Workshop "Inverse Problems and Approximation Techniques in Planetary Sciences" in Sophia Antipolis, Modelling Earth's gravitational potential with a learned best basis
- 10.2018: Workshop "Geodätische Woche" in Frankfurt a.M., Wie lernt man eine geeignete Basis für die Gravitationsfeldmodellierung?
- 04.2019: Conference "European Geosciences Union General Assembly 2019" in Vienna, Advancements for gravity field modelling using learning techniques
- 07.2019: Conference "Applied Inverse Problems Conference 2019" in Grenoble, invited talk in minisymposium "Inverse Problems in Planetary Sciences and Medical Imaging", A dictionary learning approach for inverse problems on the sphere