Dynamical Programming for off-the-grid dynamic Inverse Problems

Duval, Vincent; Tovey, Robert

doi:10.48550/arxiv.2112.11378

Cited by 2 publications

(3 citation statements)

References 17 publications

(48 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, the verification of hypotheses (B1)-(B5), necessary for ensuring fast convergence of Algorithm 1, is non-trivial and is left to future work. We remark that an implementable version of Algorithm 1 for solving (4.27) under specific choices of the fidelity term F and the operator K has been recently proposed in [16] (see also [38]). We refer to these papers for more details about the practical implementation and the modifications needed to deal with time dependent measurement operators.…”

Section: Dynamic Inverse Problems Regularized With the Benamou-brenie...mentioning

confidence: 99%

“…Together with further acceleration strategies, this method is shown to be computationally feasible and very accurate for the task of tracking several dynamic sources in presence of high noise and severe spatial undersampling. In [38], the authors proposed to speed up the algorithm in [16] by considering inexact subproblems (3.1) and solving them using known algorithms for computing shortest paths on directed acyclic graphs.…”

Section: Dynamic Inverse Problems Regularized With the Benamou-brenie...mentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic linear convergence of fully-corrective generalized conditional gradient methods

et al. 2023

View full text Add to dashboard Cite

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.

show abstract

Section: Dynamic Inverse Problems Regularized With the Benamou-brenie...mentioning

confidence: 99%

Section: Dynamic Inverse Problems Regularized With the Benamou-brenie...mentioning

confidence: 99%

Asymptotic linear convergence of fully-corrective generalized conditional gradient methods

et al. 2023

View full text Add to dashboard Cite

show abstract

“…Moreover, it allows to devise accelerated generalized conditional gradient algorithms [8], i.e. infinite dimensional versions of the classical Frank-Wolfe algorithm [14,13,18,31] that are based on the iterative construction of linear combination of extremal points, converging to a solution of the minimization problem [20,7,16,6]. These methods and algorithms are applicable to Problem (1) and they allow to formulate an optimization procedure that does not entail an inner minimization anymore.…”

Section: Introductionmentioning

confidence: 99%

Extremal points and sparse optimization for generalized Kantorovich-Rubinstein norms

Carioni¹,

Iglesias²

2022

Preprint

View full text Add to dashboard Cite

A precise characterization of the extremal points of sublevel sets of nonsmooth penalties provides both detailed information about minimizers, and optimality conditions in general classes of minimization problems involving them. Moreover, it enables the application of accelerated generalized conditional gradient methods for their efficient solution. In this manuscript, this program is adapted to the minimization of a smooth convex fidelity term which is augmented with an unbalanced transport regularization term given in the form of a generalized Kantorovich-Rubinstein norm for Radon measures. More precisely, we show that the extremal points associated to the latter are given by all Dirac delta functionals supported in the spatial domain as well as certain dipoles, i.e., pairs of Diracs with the same mass but with different signs. Subsequently, this characterization is used to derive precise first-order optimality conditions as well as an efficient solution algorithm for which linear convergence is proved under natural assumptions. This behaviour is also reflected in numerical examples for a model problem.F (Kµ) + α µ TV ,

show abstract

Dynamical Programming for off-the-grid dynamic Inverse Problems

Cited by 2 publications

References 17 publications

Asymptotic linear convergence of fully-corrective generalized conditional gradient methods

Asymptotic linear convergence of fully-corrective generalized conditional gradient methods

Extremal points and sparse optimization for generalized Kantorovich-Rubinstein norms

Contact Info

Product

Resources

About