An optimal transport approach for solving dynamic inverse problems in spaces of measures

Bredies, Kristian; Fanzon, Silvio

doi:10.1051/m2an/2020056

Cited by 16 publications

(62 citation statements)

References 65 publications

(138 reference statements)

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“…Therefore, using that ρt is concentrated on Ω, we conclude that σ(A) = 0, showing that σ is concentrated on Γ v (Ω). Finally, ( 16) implies (15) since ρt is supported in Ω and it coincides with ρ t in Ω. Also Γ v (Ω) = Γ v (Ω) by definition of v, thus concluding the proof.…”

Section: The Superposition Principlementioning

confidence: 54%

“…Since v(t, •) belongs to L 2 ρt (Ω; R d ) for a.e. t ∈ (0, 1), by [9, Theorem 3.6.1], (15) and the definition of σ 1 , we get…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…For α > 0 and β > 0, we regularize the above inverse problem by means of the energy J α,β defined in (10), following the approach in [15]. In practice, upon introducing the space…”

Section: Application To Sparse Representation For Inverse Problems With Optimal Transport Regularizationmentioning

confidence: 99%

“…Moreover, it motivated recent developments in unbalanced optimal transport theory [20,21,32,33], that is, when the marginals are arbitrary positive measures. Finally, as the Benamou-Brenier energy provides a description of the optimal flow of the transported mass at each time t, which is a valuable information in applications, it was recently employed as a regularizer for variational inverse problems [13,15,28,34,49] (see also a forthcoming paper by Bredies, Carioni, Fanzon and Walter). The goal of this paper is to characterize the extremal points of the unit ball of the Benamou-Brenier energy B at (3), and of a coercive version of it, which is obtained by adding the total variation of ρ to B.…”

Section: Introductionmentioning

confidence: 99%

“…In the last 30 years, great advances in the understanding of the underlying theory have been achieved [3,45,48]. However, only recently these techniques are starting to be applied in order to solve computational problems in a great variety of fields, with logistic problems [8,[17][18][19], crowd dynamics [36,37], image processing [28,34,38,40,43,46,47], inverse problems [15,31] and machine learning [5,26,27,39,44,51] being a few examples.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Bredies

Carioni

Fanzon

et al. 2021

Bull. London Math. Soc.

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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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Section: The Superposition Principlementioning

confidence: 54%

“…Since v(t, •) belongs to L 2 ρt (Ω; R d ) for a.e. t ∈ (0, 1), by [9, Theorem 3.6.1], (15) and the definition of σ 1 , we get…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…For α > 0 and β > 0, we regularize the above inverse problem by means of the energy J α,β defined in (10), following the approach in [15]. In practice, upon introducing the space…”

Section: Application To Sparse Representation For Inverse Problems With Optimal Transport Regularizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Bredies

Carioni

Fanzon

et al. 2021

Bull. London Math. Soc.

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Asymptotic linear convergence of fully-corrective generalized conditional gradient methods

et al. 2023

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We propose a fully-corrective generalized conditional gradient method (FC-GCG) for the minimization of the sum of a smooth, convex loss function and a convex one-homogeneous regularizer over a Banach space. The algorithm relies on the mutual update of a finite set $$\mathcal {A}_k$$ A k of extremal points of the unit ball of the regularizer and of an iterate $$u_k \in {\text {cone}}(\mathcal {A}_k)$$ u k ∈ cone ( A k ) . Each iteration requires the solution of one linear problem to update $$\mathcal {A}_k$$ A k and of one finite dimensional convex minimization problem to update the iterate. Under standard hypotheses on the minimization problem we show that the algorithm converges sublinearly to a solution. Subsequently, imposing additional assumptions on the associated dual variables, this is improved to a linear rate of convergence. The proof of both results relies on two key observations: First, we prove the equivalence of the considered problem to the minimization of a lifted functional over a particular space of Radon measures using Choquet’s theorem. Second, the FC-GCG algorithm is connected to a Primal-Dual-Active-Point method on the lifted problem for which we finally derive the desired convergence rates.

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A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization

et al. 2022

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We develop a dynamic generalized conditional gradient method (DGCG) for dynamic inverse problems with optimal transport regularization. We consider the framework introduced in Bredies and Fanzon (ESAIM: M2AN 54:2351–2382, 2020), where the objective functional is comprised of a fidelity term, penalizing the pointwise in time discrepancy between the observation and the unknown in time-varying Hilbert spaces, and a regularizer keeping track of the dynamics, given by the Benamou–Brenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the Benamou–Brenier energy (Bredies et al. in Bull Lond Math Soc 53(5):1436–1452, 2021), we define the atoms of the problem as measures concentrated on absolutely continuous curves in the domain. We propose a dynamic generalization of a conditional gradient method that consists of iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimizing the coefficients in the resulting linear combination. We prove that the method converges with a sublinear rate to a minimizer of the objective functional. Additionally, we propose heuristic strategies and acceleration steps that allow to implement the algorithm efficiently. Finally, we provide numerical examples that demonstrate the effectiveness of our algorithm and model in reconstructing heavily undersampled dynamic data, together with the presence of noise.

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An optimal transport approach for solving dynamic inverse problems in spaces of measures

Cited by 16 publications

References 65 publications

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

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

Asymptotic linear convergence of fully-corrective generalized conditional gradient methods

A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization

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