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.

Dynamical Programming for off-the-grid dynamic Inverse Problems

“…A key analytical tool in related prior works (c.f. Bredies et al [2021], Bredies and Fanzon [2020], Bredies et al [2020a,b]) is a mapping between measures ρ ∈ M([0, 1] × Ω) which satisfy the continuity equation, and measures on (weighted) paths σ ∈ M(Γ 0 ), making some structures become more apparent. We will now recap and expand on previous results.…”

“…The purpose of this section is to motivate our framework using a specific example which has been studied in the seminal work of Bredies et al [2021Bredies et al [ , 2020b. Following Bredies et al [2021Bredies et al [ , 2020b, we define for α, β > 0 the Benamou-Brenier penalty W :…”

“…The signal-processing task of dynamical super-resolution involves retrieving fine-scale features, in space and time, from a signal which evolves over time. A convex variational model was recently proposed for such tasks using Optimal Transport (OT) to regularise the associated inverse problem [Bredies et al, 2021, Bredies andFanzon, 2020]. This new approach allows the decomposition of a signal into a finite sum of smooth curves, for example to track the centers of multiple particles in time with smooth trajectories.…”

“…These classes are referred to as balanced or unbalanced problems respectively, mathematically encoded in the choice of OT cost. Current literature provides analysis for the balanced Benamou-Brenier (BB) [Bredies et al, 2021] and the unbalanced Wasserstein-Fisher-Rao (WFR) [Bredies et al, 2020a] energies, both are shown to reconstruct data into a finite number of smooth curves with constant or smoothly varying mass. Initial numerical experiments for the Benamou-Brenier model have also been carried out showing great promise [Bredies et al, 2020b], however current methods are too slow for large-scale applications.…”

“…In the particular case of dynamical super-resolution using optimal transport regularisation, those extreme points are described by the key results of Bredies et al [2021Bredies et al [ , 2020a (see also Ambrosio et al [2008]) as measures supported on the above-mentioned curves γ = (h, ξ) : [0, 1] → R × Ω. Even though only a small subset of L ∞ ([0, 1] ; R × Ω) is involved, optimising over such a set is a challenging task that is addressed in Bredies et al [2020b] in the case of balanced optimal transport.…”