The star geometry of regularizer learning

Music City Center 

Regularization is a classical approach for statistical inference and inverse problems. Modern data-driven approaches parameterize regularizers via deep neural networks and learn task-dependent regularizers, showcasing impressive empirical performance. In particular, critic-based regularizers integrate information about the measurements and ground-truth data in an unsupervised loss function, and a regularizer is learned that attributes low values to likely data and high values to unlikely data. However, there is little theory about the structure of regularizers learned via this process. In this talk, we will study this critic-based approach for a particular family of regularizers: Minkowski functionals of star-shaped bodies. By leveraging tools from star geometry and dual Brunn-Minkowski theory, we derive exact expressions for the optimal regularizer in certain cases. We also study the relationship between this class of regularizers and neural network architectures, as well as illustrate their potential practical application with empirical studies.

Regularizer learning

Star Bodies