TITLE: Minimization in spaces of measures: No-gap second order conditions and fast algorithms

ABSTRACT: Across a wide variety of mathematical problems, sparsity of the sought solutions, i.e. their representation as a superposition of few structured atoms, is a core principle. In this context, a pivotal role is played by minimization problems over spaces of Radon measures yielding solutions comprising finitely many Dirac-Delta peaks. In many applications, these problems arise naturally as a lifting or relaxation of nonlinear or discrete problems, thus trading of the complexity of the latter for the necessity to work in infinite-dimensional decision spaces as well as potential nonsmoothness. For example, mean-field approaches in certain machine learning problems have provided new theoretical insights and offer an elegant way towards the study of large-data limits as well as infinite network depth and width considerations.

In this talk, we present no-gap second order conditions for minimization problems in spaces of measures and discuss the fundamental role of the former in the design and analysis of solution algorithms as well as their inherent link to unbalanced optimal transport.