Data-driven and problem-oriented choice of the regularization space

Regularization is an approach to approximate reconstruction of the unknown functional dependency from available noisy data. This approach is often based on a compromise between the attempt to fit given data and the desire to reduce complexity of a data fitter. Starting from the pioneering work of Tikhonov and Wahba, Kimeldorf in the the mid-sixties a huge body of the regularization theory has been built around the issue of choosing the regularization parameter, which is just a one- number choice. At the same time, the most recent trend in the regularization theory has led to a new line of research of the adaptive choice of the regularization space, with still many open questions. The current project aims at comprehensive theoretical analysis of this issue and extensive numerical studies. In particular, two research directions will be explored. One of them leads to the multi-penalty regularization (MPR), where only the first theoretically justified results on adaptive selection of multiple regularization parameters have been obtained. Since the idea of a MPR usage is attractive as it opens a possibility of combining several approximation tasks such as predictions by interpolation and by extrapolation, one argues that under some assumptions, regularization in an adaptively chosen space can be reduced to a multi-penalty regularization with a component-wise penalization. Another research direction appears on the border between regularization theory and meta-learning. In rough terms, a meta-learning based regularization means that the instances of a regularization method are chosen from experience with this method in similar applications. Such frame covers several recently proposed approaches, where only a single instance is extracted from the experience. But it seems to be more promising to use the experience for finding a rule for choosing regularization instances in dependence on features (meta-features) of a particular application. It appears that such approach has not been systematically studied so far in the regularization theory and the current project aims to shed the light on this promising but as of yet not researched area. Both above mentioned project directions may, in the future, form one of mainstreams in the regularization theory and can play a fundamental role for some practical applications, such as geomathematics and diabetes technology.

Regularization of ill-posed problems is based on a compromise between the attempt to fit given data and desire to reduce complexity of a data fitter, which is usually constructed in a priori chosen space. In contrast, the Project P25424-N26 was aimed at studying the possibility of improving the regularization performance by adaptive adjustment of the space, in which data fitting is executed.The main efforts were concentrated on two forms of the regularization space adjustment: multiple penalty regularization and aggregation of several regularized data fitters. Sound results for both these forms have been obtained by the project team and their usefulness demonstrated by applications in Geomathematics, Machine learning and Diabetes technology.One of the project achievements is a new regularization approach consisting in the aggregation of different regularized approximants by means of the so-called linear functional strategy. This approach has been directly applied to the problem of the prediction of hypoglycemia in diabetes patients, and previously known clinical prediction rules have been aggregated according to the developed methodology. The resulting prediction algorithm has been implemented in the form of a diabetic smartphone app exhibiting a reliable performance and the potential for everyday use by any patient who does self-monitoring of blood glucose. This research has been performed in cooperation with clinical partners and colleagues from our other project AMMODIT funded within EU Horizon 2020 Programme.