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Research topics

The research group "Mathematical Statistics & Machine Learning" headed by Prof. Dr. Alexandra Carpentier is part of the Institute of Mathematics, University Potsdam. We are focusing on problems in mathematical statistics and machine learning:

- We are intereserted in unsupervised learning and on statistical-computational trade-off that arise in related fields (clustering, ranking, seriation...). We aim at crafting and analysing polynomial-time algorithms for these problems. We are also interested in characterising the fundamental limits of learning through polynomial-time algorithms in these settings.
- Another main research interest of our group is composite-composite testing problems, where we aim at understanding which properties of the statistical hypotheses drive the minimax separation rate between the hypotheses so that a non-trivial test exists. Specific examples of such questions are crucial for uncertainty quantification in high and infinite dimensional models. We also study problems of adaptive inference in different settings, e.g. for matrix completion and extreme value theory.
- More from the machine learning side, a main research interest of our group is sequential sampling, and in particular sequential and adaptive sampling problems related to the bandit problem. We study active strategies and their theoretical performance, and are interested in constructing algorithms that are efficient in a minimax optimal sense. We consider application to neuroscience and recommender systems.
- We combine our domains of research in order to consider sequential learning on complex systems in an unsupervised learning context, and focus in particular on the problem of adapting sequential strategy to the model generating the data, using elements from adaptive inference and uncertainty quantification.

### Keywords :

High or Infinite-Dimensional Statistical Inference, Unsupervised Learning (clustering, ranking, seriation...), Sequential Sampling and Bandit Algorithms, Anomaly Detection, Applications in Digital Engineering and Neuroscience.

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Third party funded projects

### ASCAI project PRCI CA 1488/4-1:

Unsupervised Learning is one of the most fundamental problem of machine learning, and more generally, of artificial intelligence. In a broad sense, it amounts to learning some unobserved latent structure over data. This structure may be of interest per se, or may serve as an important stepping stone integrated in a complex data analysis pipe-line - since large amounts of unlabeled data are more common than costly labeled data. Arguably, one the cornerstones of unsupervised learning is clustering, where the aim is to recover a partition of the data into homogeneous groups. Beside vanilla clustering, unsupervised learning encompasses a large variety of related other problems such as hierarchical clustering, where the group structure is more complex and reveals both the backbone and fine-grain organization of the data, segmentation where the shape of the clusters is constrained by side information, or ranking or seriation problems where where no actual cluster structure exists, but where there is some implicit ordering between the data. All these problems have already found countless applications and interest in these methods is even strengthening due to the amount of available unlabelled data. We can for instance cite crowdsourcing - where individuals answer to a subset of questions, and where, depending on the context, one might want to e.g. cluster them depending on their field of expertise, rank them depending on their performances, or seriate them depending on their affinities. Such problems are extremely relevant for recommender systems - where individuals are users, and questions are items - and for social network analyses.

The analysis of unsupervised learning procedures has a long history that takes its roots both in the computer science and in mathematical communities. In response to recent bridges between these two communities, groundbreaking advances have been made in the theoretical foundations of vanilla clustering. We believe that these recent advances hold the key for deep impacts on the broader field of unsupervised learning because of the pervasive nature of clustering. In this proposal, we first aim at propagating these recent ground-breaking advances in vanilla clustering to problems where the latent structure is either more complex or more constrained. We will consider problems of increasing latent structure complexity - starting from hierarchical clustering and heading toward ranking, seriation, and segmentation - and propose new algorithms that will build on each other, focusing on the interfaces between these problems. As a result, we expect to provide new methods that are valid under weaker assumptions in comparison to what is usually done - e.g. parametric assumptions - while being also able to adapt to the unknown intrinsic difficulty of the problem.

Moreover, many modern unsupervised learning applications are essentially of an online nature - and sometimes decisions have to be made sequentially on top of that. For instance, consider a recommender systems that sequentially recommends items to users. In this context where sequential, active recommendations are made, it is important to leverage the latent structure underlying the individuals. While both the fields of unsupervised learning, and sequential, active learning, are thriving, research at the crossroad has been conducted mostly separately by each community - leading to procedures that can be improved. A second aim of this proposal will then be to bring together the fields of unsupervised learning and active learning, in order to propose new algorithms that are more efficient at leveraging sequentially the unknown latent structure. We will consider the same unsupervised learning problems as in the batch learning side of the proposal. We will focus on developing algorithms that fully take advantage of new advances in clustering, and of our own future work in batch learning.

### Emmy Noether project MuSyAD on Anomaly Detection CA 1488/1-1 (DFG):

Alexandra Carpentier is an Emmy Noether research group leader for the project MuSyAD (CA 1488/1-1) "Anomaly Detection in a Multi-System Setting: Theoretical and Computational Objective", funded by the Deutsche Foschungsgemeinschaft (DFG, German Research Foundation). Andrea Locatelli and Maurilio Gutzeit are also funded by this project. Our project is on the topic of anomaly detection. 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 we propose to study a more complex scenario, which will be referred to as multi system anomaly detection. In this setting, reference systems can have a variety of normal behaviours, and moreover, there are many systems under the surveillance of the monitor, 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.

### FOR 5381 Mathematical Statistics in the Information Age (DFG):

The group is also funded by the DFG on the Research Unit__FOR 5381__** "**Mathematical Statistics in the Information Age - Statistical Efficiency and Computational Tractability".

SFB 1294 Data Assimilation (DFG), projet A03:

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.

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**FORMER PROJECTS**

### GRK 2297 MathCore (DFG):

The group is also funded by the Deutsche Foschungsgemeinschaft (DFG, German Research Foundation) on the GRK 2297 MathCoRe on “Mathematical Complexity Reduction" 314838170, GRK 2297 MathCoRe. 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.

### Sachsen-Anhalt WISSENSCHAFT Spitzenforschung/Synergien Project RE-BCI:

The project RE-BCI was awarded in the beginning of 2020 by the Land Sachsen Anhalt, more pre-

cisely 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.

### GRK 2433 DAEDALUS (DFG):

The group is also funded by the Deutsche Foschungsgemeinschaft (DFG, German Research Foundation) on the GRK 2433 DAEDALUS. 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.

### Amazon Postdoctoral program:

The group is also funded by Amazon Research postdoctoral program. Dr. Claire Vernade is the concerned postdoc and is sharing her time between Amazon Research in Berlin and the OvGU.