A Mathematical Journey of Regularization on Measures

In this presentation, we will take a tour of the path of regularization on measures. We will use an example to revisit recent developments in this tool at the intersection of statistics, learning, and optimization. Along the way, we will encounter various species such as the dual of a convex program with its subgradient, empirical process concentration with its "golfing scheme," kernel functions with their Hilbertian structures, stochastic gradient descent with particles, and beautiful weather days with an optimal transport distance called partial displacement. The landscapes traversed will lead us to discuss applications in unsupervised learning (deep or not), super-resolution, tensor processing or even quadrature. We will avoid technical details on sensitive topics, but any questions about these aspects are welcome. No technical equipment is required; a regular practice of indulgence is sufficient.