Applications of weak transport theory

Backhoff-Veraguas, Julio; Pammer, Gudmund

doi:10.48550/arxiv.2003.05338

Cited by 2 publications

(4 citation statements)

References 45 publications

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“…The WOTUK problem (8) admits dual formulations that are similar to the dual WOT formulations (5) and (6). The main difference with the results of subsection 2.3 lies in the fact that P(Y) is replaced by M (Y) and that conv(Y) is replaced by cone(Y).…”

Section: Dualitymentioning

confidence: 99%

“…We can construct dual minimizers for the barycentric WOT (6) and conical WOTUK (12) from a solution ϕ of ( 14) and ( 15) respectively, using the results given in ( 7) and ( 13) respectively. In the discrete setting, these linear programs respectively write: Since we may be interested in differentiating those functions ψ , we rather compute the dual problems of the above linear programs (see a proof in the Appendix B.2): ψ : z → max λ∈R q ,µ∈R ∀j, λ,yj +µ≤ϕ j λ, z + µ and ψ : z → max λ∈R q ∀j, λ,yj ≤ϕ j λ, z .…”

Section: The Dual Problemsmentioning

confidence: 99%

“…These variants of OT are unable to capture the aggregation property that plays a crucial role in some applications [6,12]. In these applications, the cost for matching points x and y does not only depends on x and y through a cost function, but also on all the other points y matched with x.…”

Section: Introductionmentioning

confidence: 99%

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Algorithms for Weak Optimal Transport with an Application to Economics

Paty¹,

Choné²,

Kramarz³

2022

Preprint

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The theory of weak optimal transport (WOT), introduced by [23], generalizes the classic Monge-Kantorovich framework by allowing the transport cost between one point and the points it is matched with to be nonlinear. In the so-called barycentric version of WOT, the cost for transporting a point x only depends on x and on the barycenter of the points it is matched with. This aggregation property of WOT is appealing in machine learning, economics and finance. Yet algorithms to compute WOT have only been developed for the special case of quadratic barycentric WOT, or depend on neural networks with no guarantee on the computed value and matching. The main difficulty lies in the transportation constraints which are costly to project onto. In this paper, we propose to use mirror descent algorithms to solve the primal and dual versions of the WOT problem. We also apply our algorithms to the variant of WOT introduced by [13] where mass is distributed from one space to another through unnormalized kernels (WOTUK). We empirically compare the solutions of WOT and WOTUK with classical OT. We illustrate our numerical methods to the economic framework of [12], namely the matching between workers and firms on labor markets.Preprint. Under review.

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Section: Dualitymentioning

confidence: 99%

Section: The Dual Problemsmentioning

confidence: 99%

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Algorithms for Weak Optimal Transport with an Application to Economics

Paty¹,

Choné²,

Kramarz³

2022

Preprint

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“…F(q) Q(dq). Weak optimal transport problems where the cost function C is of barycentric-type have been recently investigated by various authors in different contexts, see Gozlan et al (2018Gozlan et al ( , 2017; Gozlan and Juillet (2020); Shu (2020); Alfonsi et al (2020); Daskalakis et al (2017); , 2020b; Backhoff-Veraguas and Pammer (2020). Typically in these papers, X was also given as R d and C(x, p) = θ(b(p) − x) for some convex θ : R d → R. Motivated by functional inequalities, this particular problem was explored by Gozlan et al (2017Gozlan et al ( , 2018; Shu (2020); Gozlan and Juillet (2020).…”

Section: Therefore We Havementioning

confidence: 99%

Weak Transport for Non-Convex Costs and Model-independence in a Fixed-Income Market

Acciaio¹,

Beiglboeck²,

Pammer³

2020

Preprint

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A. We consider a model-independent pricing problem in a fixed-income market and show that it leads to a weak optimal transport problem as introduced by Gozlan et al. We use this to characterize the extremal models for the pricing of caplets on the spot rate and to establish a first robust super-replication result that is applicable to fixed-income markets.Notably, the weak transport problem exhibits a cost function which is non-convex and thus not covered by the standard assumptions of the theory. In an independent section, we establish that weak transport problems for general costs can be reduced to equivalent problems that do satisfy the convexity assumption, extending the scope of weak transport theory. This part could be of its own interest independent of our financial application, and is accessible to readers who are not familiar with mathematical finance notations.

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Applications of weak transport theory

Cited by 2 publications

References 45 publications

Algorithms for Weak Optimal Transport with an Application to Economics

Algorithms for Weak Optimal Transport with an Application to Economics

Weak Transport for Non-Convex Costs and Model-independence in a Fixed-Income Market

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