Special CMX Seminar

Across many tasks in data science, it is necessary to estimate a signal from corrupted measurements. Perhaps the most pervasive and commonly used technique to address such problems is variational regularization. This consists of solving an optimization problem where one must minimize the sum of a data fidelity term and a regularizer, a penalty term chosen to encourage certain structure in solutions. While there is a suite of regularizers one could choose from, we currently lack a systematic understanding, from a modeling perspective, of what types of geometries should be preferred in a regularizer for a given data source. In particular, given a data distribution, what is the "optimal" regularizer for such data? Moreover, what aspects of the data govern whether the regularizer enjoys certain properties, such as convexity? Using ideas from star geometry, Brunn–Minkowski theory, and variational analysis, I will show that one can characterize the optimal regularizer for a given distribution and establish conditions under which this optimal regularizer is convex. I will then discuss two recent extensions of this theory: a distributionally robust framework for learning regularizers under uncertainty, and a raywise transport perspective showing how direction-preserving transformations can modify a data distribution so as to realize a prescribed optimal regularizer.