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Description:
Optimization methods in the space of measures have seen a dramatic increase in popularity in recent years. This is largely due to two factors: their connections with Optimal Transport (OT) and Wasserstein Gradient Flows, and their applicability to solve inverse problems and design infinite-dimensional machine learning algorithms.
The goal of this project is the analysis of dynamic optimization problems for time-dependent measures, where motion is governed by the penalization of OT energies. We plan to adopt a sparse perspective, analyzing how collections of trajectories can be employed to approximate dynamic distributions and how gradient descent methods in the space of curves can be used to achieve this goal. A full understanding of these dynamic problems and their efficient optimization is essential to address a wide class of inverse problems and machine learning methods. In particular, applications to single-particle tracking (SPT) for fluorescence microscopy and neuralODEs training will be considered.