The Maximum Entropy of the Mean (MEM) is a general approach to statistical estimation dating back to ideas of Jaynes in 1957.

We begin by exploring MEM as a regularization method in the blind deblurring and denoising of images containing some form of symbology; for example, QR bar codes.

We then address the MEM in far more generality, focusing on the relation between the MEM function (which is defined via an

infinite-dimensional variational problem) and

the Cramer rate function, famous in large deviation theory.

In doing so, we see how, for a variety of known prior distributions, the MEM function can be explicitly computed and efficiently implemented in a variety of models.

Finally, we return to the MEM as a method to solve linear inverse problems via an empirical measure approximation of a prior distribution. We provide both theoretical results and examples for denoising with data-driven priors.

This talk consists of work with Tim Hoheisel (McGill).

As we will discuss, various parts of the talk comprise key collaborators:

Former McGill undergraduates Gabriel Rioux (PhD student at Cornell), Christopher Scarvelis (PhD student at MIT), and Ariel Goodwin (PhD student at Cornell);

Former McGill postdoc Yakov Vaisbourd;

Current McGill PhD student Matt King-Roskamp;

And Pierre Marechal (Toulouse), and Carola-Bibiane Schonlieb (Cambridge).

Bio: Rustum Choksi is Professor and Department Chair in the Department of Mathematics and Statistics and McGill University.

He completed his undergraduate degree at the University of Toronto in 1989 and went on to complete his Ph.D. at Brown University in 1994. After holding post-doctoral positions at the Center for Nonlinear Analysis, Carnegie Mellon University and the Courant Institute, New York University, he was a faculty member with the Department of Mathematics at Simon Fraser University. Since 2010, he has been at McGill University. He views himself as an applied mathematician, and, broadly speaking, his research lies in the study and application of variational problems in the physical sciences, data sciences, and in geometry from the perspective of the calculus of variations, partial differential equations, optimization, and scientific computing.