Functions describe the dependence of a target variable upon one or several
independent variables. These variables can be all sorts of measurable
quantities like the maximum temperature in Vienna for tomorrow based on
various weather data from today or the probability for skin cancer based on a
picture of the skin. The precise dependency between the influential and the
target quantity are usually unknown. We therefore want to learn the
dependency with best possible accuracy, or in other words: We want to
approximate the function. Based on our approximation, we can then make
predictions of the target value in the future.
To learn the dependency we have access to a certain kind of data. In some
cases, we may collect the data actively and perform measurements that tell us
as much as possible about the function. In other cases, our influence on the
choice of the measurements is restricted or we have to cope with the data that
is already at hand. This project deals with the question to what extent these
restrictions lead to a worse prediction. For example, there are surprisingly
many cases, where randomly obtained data is almost as good as data that is
obtained from complicated and carefully selected measurements. This is one of
the phenomena which we seek to understand.
However, instead of specific examples like weather forecasts, we investigate
the above research questions for abstract classes of functions that are as
general as possible, so that the results are useful for a large variety of
applications. The project therefore deals with the mathematical foundations
and the models behind these phenomena.