Humboldt-Universität zu Berlin - Mathematisch-Naturwissen­schaft­liche Fakultät - International Research Training Group 1740


Project A1: Coupled dynamical systems and applications to semi-supervised classification task

Research Team: E. Macau (INPE), M. Quiles (UNIFESP), S. Yanchuk (TUB), L. Zhao (USP)

Outline: Networks of dynamical systems appear as mathematical models in a variety of applications including neuroscience, laser dynamics, mechanics, earth science, etc. In this project, we would like to study coupled dynamical systems appearing in semi-supervised learning techniques, based on the principles borrowed from the coordinated behavior of certain living species like flocking of birds or schooling of fishes. Each item in the given set of data is treated as an individual in the flock. The individuals are interacting, and as a result, they form several ''flocks'' or clusters. The aim of the project is to study the basic properties of the class of dynamical systems involved in such learning scheme, e.g. dynamical stability, convergence rates, possible multistabilities, which potentially lead to distinct classification results. On the basis of the obtained results, a new or optimized semi-supervised classification techniques will be proposed.

Research Topic: The mathematical models involved in the above mentioned classification techniques are interacting dynamical systems, such as coupled ordinary differential equations or discrete maps with non-smoothness. The dimensionality of the model is proportional to the amount of data to classify and may become very high. This fact as well as the non-smoothness of the governing dynamical rule makes the problem challenging from the analytical as well as numerical point of view. Although the examples from have evidenced the succesful applicability of these methods, the study of the dynamics is necessary in order to clarify the generality of the method, to optimize its convergence rate and stability of the cluster states. Particular attention will be paid to the possible multistability, which can lead to the variations in classification results. Within the German group, we will study the mathematical models involved in the classification techniques using the expertize in coupled dynamical systems. In addition, a basic research on synchronization in delay-coupled networks will be continued in cooperation with T. Pereira.