Functional lifting, direct approaches and applications of nonconvex optimization in computer vision

Optimization problems are ubiquitous in computer vision, machine learning, economics and basically any field in the domain of natural and engineering sciences. Developments in optimization theory and algorithms have hence always been tightly interconnected to the problems arising from practical applications. This work follows the same path studying and developing various optimization techniques and models depending on the problem at hand. Fruitful optimization theory has been developed around convex problems and applied in computer vision tasks. In this work we extensively study theoretical properties of functional lifting techniques which make even nonconvex problems amenable to the tools from convex optimization by increasing the dimensionality of the original problem formulations. Furthermore we propose enhancements making existing approaches either more faithful or more efficient. Following the idea of increasing the dimensionality we study the effects of reparameterizations and neural network models with adaptive expressivity in the domain of machine learning. For linear inverse problems we devise an optimization approach combining data driven priors with provable convergence guarantees. Finally, we propose an optimization based approach for solving high-dimensional problems arising from attacks on data security in the application of federated machine learning.