On the extremal points of the ball of the Benamou-Brenier energy

Abstract

In this paper we characterize the extremal points of the unit ball of the Benamou--Brenier energy and of a coercive generalization of it, both subjected to the homogeneous continuity equation constraint. We prove that extremal points consist of pairs of measures concentrated on absolutely continuous curves which are characteristics of the continuity equation. Then, we apply this result to provide a representation formula for sparse solutions of dynamic inverse problems with finite dimensional data and optimal-transport based regularization.

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Cited by in corpus (5)

• A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization
• Asymptotic linear convergence of fully-corrective generalized conditional gradient methods
• Towards a mathematical theory of trajectory inference
• A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients
• Extremal points and sparse optimization for generalized Kantorovich-Rubinstein norms