Variational Regularization in Inverse Problems and Machine Learning
It provides a theoretical framework connecting regularization in inverse problems and machine learning, which is incremental but useful for researchers in both domains.
This paper reviews variational regularization methods for inverse problems and machine learning, establishing connections between the two fields and deriving new links between Bregman distance error estimates and generalization errors.
This paper discusses basic results and recent developments on variational regularization methods, as developed for inverse problems. In a typical setup we review basic properties needed to obtain a convergent regularization scheme and further discuss the derivation of quantitative estimates respectively needed ingredients such as Bregman distances for convex functionals. In addition to the approach developed for inverse problems we will also discuss variational regularization in machine learning and work out some connections to the classical regularization theory. In particular we will discuss a reinterpretation of machine learning problems in the framework of regularization theory and a reinterpretation of variational methods for inverse problems in the framework of risk minimization. Moreover, we establish some previously unknown connections between error estimates in Bregman distances and generalization errors.