Projects
Current projects
Mathematical Complexity Reduction
Duration: 01.04.2017 bis 31.03.2026
In the context of the proposed RTG we understand complexity as an intrinsic property that makes
it difficult to determine an appropriate mathematical representation of a real world problem, to assess the fundamental structures and properties of mathematical objects, and to algorithmically solve a given mathematical problem. By complexity reduction we refer to all approaches that help to overcome these difficulties in a systematic way and to achieve the aforementioned goals more efficiently.
For many mathematical tasks, approximation and dimension reduction are the most important tools to obtain
a simpler representation and computational speedups. We see complexity reduction in a more general way and
will also, e.g., investigate liftings to higher-dimensional spaces and consider the costs of data observation.
Our research goals are the development of cross-disciplinary mathematical theory and methods for complexity
reduction and the identification of relevant problem classes and effective exploitation of their structures.
We aim at a comprehensive teaching and research program based on geometric, algebraic, stochastic, and
analytic approaches, complemented by efficient numerical and computational implementations. In order to
ensure the success of our doctoral students, they will participate in a tailored structured study program. It will
contain training units in form of compact courses and weekly seminars, and encourage early integration into the
scientific community and networking. We expect that the RTG will also serve as a catalyst for a dissemination
of these successful practices within the Faculty of Mathematics and improve the gender situation.
Complexity reduction is a fundamental aspect of the scientific backgrounds of the principal investigators.
The combination of expertise from different areas of mathematics gives the RTG a unique profile, with high
chances for scientific breakthroughs. The RTG will be linked to two faculties, a Max Planck Institute, and
several national and international research activities in different scientific communities.
The students of the RTG will be trained to become proficient in a breadth of mathematical methods, and
thus be ready to cope with challenging tasks in particular in cross-disciplinary research teams. We expect an
impact both in terms of research successes, and in the education of the next generation of leading scientists in
academia and industry.
Completed projects
Risk Estimation for Brain-Computer Interfaces
Duration: 01.06.2020 bis 30.06.2022
The project RE-BCI was awarded in the beginning of 2020 by the Land Sachsen Anhalt, more precisely by the Sachsen-Anhalt WISSENSCHAFT Spitzenforschung/Synergien. The objective of RE-BCI is to prepare preliminary results supporting the BCI (Brain-Computer Interfaces, i.e. a technology for connecting a human user with a computer through the lectrical impulses emitted by her/his brain) application to shared authority situations.
Participation in the SFB 1294 on Data Assimilation in Potsdam
Duration: 01.11.2018 bis 30.11.2021
The group is also funded by the Deutsche Foschungsgemeinschaft (DFG, German Research Foundation) on the SFB 1294 Data Assimilationon "Data Assimilation - The seamless integration of data and models" on Project A03 together with Prof. Gilles Blanchard.
This project is concerned with the problem of learning sequentially, adaptively and in partial information on an uncertain environment. In this setting, the learner collects sequentially and actively the data, which is not available before-hand in a batch form. The process is as follows: at each time t, the learner chooses an action and receives a data point, that depends on the performed action. The learner collects data in order to learn the system, but also to achieve a goal (characterized by an objective function) that depends on the application. In this project, we will aim at solving this problem under general objective functions, and dependency in the data collecting process - exploring variations of the so-called bandit setting which corresponds to this problem with a specific objective function.
As a motivating example, consider the problem of sequential and active attention detection through an eye tracker. A human user is looking at a screen, and the objective of an automatized monitor (learner) is to identify through an eye tracker zones of this screen where the user is not paying sufficient attention. In order to do so, the monitor is allowed at each time t to flash a small zone a t in the screen, e.g. light a pixel (action), and the eye tracker detects through the eye movement if the user has observed this flash. Ideally the monitor should focus on these difficult zones and flash more often there (i.e. choose more often specific actions corresponding to less identified zones). Therefore, sequential and adaptive learning methods are expected to improve the performances of the monitor.
Minimax testing rates in linear regression
Duration: 01.01.2019 bis 31.10.2021
In this project we focus on finding the minimax testing rates in l_2 norm for the linear regression model. We also investigate the problem of estimating optimally the l_2 norm for the parameter. We close some gaps in linear regression.
Teilnahme an dem GK Daedalus 2433 mit der TU Berlin
Duration: 01.09.2018 bis 31.10.2021
The main goal of DAEDALUS is the analysis of the interplay between incorporation of data and differential equation-based modeling, which is one of the key problems in model-based research of the 21th century. DAEDALUS focuses both on theoretical insights and on applications in life sciences (brain-computer interfaces and biochemistry) as well as in fluid dynamics. The projects cover a scientific range from machine learning, mathematical theory of model reduction and uncertainty quantification to respective applications in turbulence theory, simulation of complex nonlinear flows as well as of molecular dynamics in chemical and biological systems. In our group, we cover mathematical statistics and machine learning aspects.
This project is in the context of Daedalus, and is concerned with uncertainty quantification in complex cases.
Minimax change point detection in high dimension
Duration: 01.01.2019 bis 01.10.2021
The objective is to establish the minimax rates for sparse change point estimation in high dimension. We want in particular to investigate in a refined way intermediary regimes. Joint project with Emmanuel Pilliat and Dr. Nicolas Verzelen.
One sample local test in the Graph model
Duration: 01.01.2019 bis 01.10.2021
In this project we aim at finding minimax rates for the problem of local testing in the graph model, in l_q norm. We focus particularly on local rates, and aim also at the multinomial tetsig model, which can be seen as a special case.
Participation in the GK 2297 Mathcore
Duration: 01.01.2019 bis 30.09.2021
The objective of this GRK is to investigate the problem of complexity reduction across the different areas of mathematics. In our group, we bring to this project some expertise on the field of sequential learning, in order to reduce the complexity of given problems by adapting the sampling strategies.
Adaptive two sample test in the density setting
Duration: 01.11.2017 bis 31.10.2020
We consider the problem of testing between two samples of (non necessarily uniform) density. While minimax signal detection in the case where the null hypothesis density is uniform is well understood, recent works in the case of multinomial distributions have highlighted the amelioration in the minimax rate that can come when considering non uniform null hypothesis density. We want to study this problem in the two sample testing case, which is significantly more complex, and extend it to smooth densities.
MuSyAD on Anomaly Detection
Duration: 01.10.2017 bis 31.08.2020
Anomaly detection is an interdisciplinary domain, borrowing elements from mathematics, computer science, and engineering. The main aim is to develop efficient techniques for detecting anomalous behaviour of systems. In the classical scenario a monitor receives data from a system and compares this data to a reference system with some single normal behaviour. Ideally no strong assumptions are made on the nature of anomalous behaviours, so the problem of anomaly detection is by essence a non parametric problem. Here I propose to study a more complex scenario, which will be referred to as multisystem anomaly detection. In this setting, reference systems can have a variety of normal behaviours, and moreover, there are many systems under the monitor s surveillance, and the monitor must allocate its resources wisely among them. In this situation new theoretical and computational challenges arise. The overall objective of this proposal is to find efficient methods to solve the problem of multi-system anomaly detection. This aim will be reached by addressing the following sub-objectives. First, we will generalise the theoretical framework of anomaly detection to the broader setting of multi-system anomaly detection. Second, multi-system anomaly detection methods will be developed, by taking ideas from the non parametric testing field and applying them to the new framework. Third, we will study optimal monitoring strategies for cases where the multiple systems cannot be monitored simultaneously. Here, it is important that the monitor allocates its resources among the systems in a way that is as efficient as possible. To this end, sequential and adaptive sampling methods that target the anomaly detection problem will be designed. Since anomaly detection is a non parametric problem, elements in the theory of non parametric confidence sets will be used. Finally, the newly developed methods will be applied to practical problems: a methodological example in extreme value theory, an econometric application for speculative bubble detection and two applications in a Brain Computer Interface framework.
Projekt on Data Assimilation
Duration: 01.10.2017 bis 31.08.2020
This project is concerned with the problem of learning sequentially, adaptively and in partial information on an uncertain environment. In this setting, the learner collects sequentially and actively the data, which is not available before-hand in a batch form. The process is as follows: at each time t, the learner chooses an action and receives a data point, that depends on the performed action. The learner collects data in order to learn the system, but also to achieve a goal (characterized by an objective function) that depends on the application. In this project, we will aim at solving this problem under general objective functions, and dependency in the data collecting process exploring variations of the so-called bandit setting which corresponds to this problem with a specific objective function.
As a motivating example, consider the problem of sequential and active attention detection through an eye tracker. A human user is looking at a screen, and the objective of an automatized monitor (learner) is to identify through an eye tracker zones of this screen where the user is not paying sufficient attention. In order to do so, the monitor is allowed at each time t to flash a small zone a t in the screen, e.g. light a pixel (action), and the eye tracker detects through the eye movement if the user has observed this flash. Ideally the monitor should focus on these difficult zones and flash more often there (i.e. choose more often specific actions corresponding to less identified zones). Therefore, sequential and adaptive learning methods are expected to improve the performances of the monitor.
Active learning for matrix completion
Duration: 01.10.2017 bis 14.06.2019
Matrix completion is an essential problem in modern machine learning, as it is e.g. important for the calibration of the recommendation systems. We consider the problem of matrix completion in the setting where the learner can choose where to sample. In this setting, it can be of interest to target more specifically parts of the matrix where it is discovered that the complexity is high (higher local rank), where the knowledge is limited (few sampled points), or where the noise is high. This project plans to consider first the problem of active learning for matrix completion when the matrix can be subdivided into block submatrices of small ranks that are known, and then in the more general case where this cannot be done.
Smoothness testing in the Sobolev sense
Duration: 01.10.2017 bis 14.06.2019
We want to develop a test to determine whether a function lying in a fixed L2-Sobolev-type ball of smoothness t, and generating a noisy signal, is in fact of a given smoothness s larger than t or not. While it is impossible to construct a uniformly consistent test for this problem on every function of smoothness t, it becomes possible if we remove a sufficiently large region of the set of functions of smoothness t. The functions that we remove are functions of smoothness strictly smaller than s, but that are very close to s-smooth functions. This problem has been considered in the case of specific Besov bodies where it is easier, and we plan to extend it to more usual Sobolev ellipsoids.