Closedness of Sum Spaces andthe Generalized Schrödinger Problem

Rüschendorf, Ludger; Thomsen, W.

doi:10.1137/s0040585x97976301

Cited by 36 publications

(28 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is thus of primary interest to find proxies for OT distances, that can provably approximate faithfully the true distance and transport plan, while offering a better computational complexity than traditional linear programming solvers such as simplex methods [18] or interior points methods [39]. This article explores the use of an entropic smoothing of the initial linear program, that was proposed initially by Schrödinger [44] (see [43,36]), and that has recently been revitalized in fields as diverse as machine learning [24], computer graphics [48] and social sciences [30].…”

Section: Introductionmentioning

confidence: 99%

Convergence of Entropic Schemes for Optimal Transport and Gradient Flows

Carlier¹,

Duval²,

Peyré³

et al. 2017

SIAM J. Math. Anal.

134

153

View full text Add to dashboard Cite

International audienceReplacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the early work of Schr\"odinger. This approach has recently been used successfully to solve optimal transport related problems in several applied fields such as imaging sciences, machine learning and social sciences. The main reason for this success is that, in contrast to linear programming solvers, the resulting algorithms are highly parallelizable and take advantage of the geometry of the computational grid (e.g. an image or a triangulated mesh). The first contribution of this article is the proof of the Γ-convergence of the entropic regularized optimal transport problem towards the Monge-Kantorovich problem for the squared Euclidean norm cost function. This implies in particular the convergence of the optimal entropic regularized transport plan towards an optimal transport plan as the entropy vanishes. Optimal transport distances are also useful to define gradient flows as a limit of implicit Euler steps according to the transportation distance. Our second contribution is a proof that implicit steps according to the entropic regularized distance converge towards the original gradient flow when both the step size and the entropic penalty vanish (in some controlled way)

show abstract

Section: Introductionmentioning

confidence: 99%

Convergence of Entropic Schemes for Optimal Transport and Gradient Flows

Carlier¹,

Duval²,

Peyré³

et al. 2017

SIAM J. Math. Anal.

134

153

View full text Add to dashboard Cite

show abstract

“…This theorem appears in [8] under a slightly different form. The proof is interesting as it provides an algorithm of determination of u and v. It is an important generalization of the Iterated Projection Fitting Procedure (see [5], [12], and [13]), which has been rediscovered and utilized many times under different names for various applied purposes: "RAS algorithm" [10], "biproportional fitting", "Sinkhorn Scaling" [4], etc. However, all these techniques and their variants can be recast as particular cases of the method described in the proof of Theorem 1.…”

Section: The Main Resultsmentioning

confidence: 99%

The Nonlinear Bernstein-Schrödinger Equation in Economics

Galichon

Kominers

Weber

2015

Lecture Notes in Computer Science

View full text Add to dashboard Cite

In this note, we will review and extend some results from our previous work [8] where we introduced a novel approach to imperfectly transferable utility and unobserved heterogeneity in tastes, based on a nonlinear generalization of the Bernstein-Schrödinger equation. We consider an assignment problem where agents from two distinct populations may form pairs, which generates utility to each agent. Utility may be transfered across partners, possibly with frictions. This general framework hence encompasses both the classic Non-Tranferable Utility model (NTU) of Gale and Shapley [6], sometimes called the "stable marriage problem", where there exists no technology to allow transfers between matched partners; and the Transferable Utility (TU) model of Becker [1] and Shapley-Shubik [14], a.k.a. "optimal assignment problem," where utility (money) is additively transferable across partners.If the NTU assumption seems natural for many markets (including school choices), TU models are more appropriate in most settings where there can be bargaining (labour and marriage markets for example). However, even in those markets, there can be transfer frictions. For example, in marriage markets, the transfers between partners might take the form of favor exchange (rather than cash), and the cost of a favor to one partner may not exactly equal the benefit to the other.In [8], we thus developped a general Imperfectly Transferable Utility model with unobserved heterogeneity, which includes as special cases the classic fully-and non-transferable utility models, but also extends to collective models, and settings with taxes on transfers, deadweight losses, and risk aversion. As we argue in the present note, the models we consider in [8] obey a particularly simple system of equations we dubbed "Nonlinear Bernstein-Schrödinger equation". The present contribution present a general result for the latter equation, and also derives several consequences. The main result is derived in Section 1. Section 2 and Section 3 consider equilibrium assignment problems with and without heterogeneity. Finally, we provide a discussion of our results in Section 4.

show abstract

“…where u i and v j are the regularized Kantorovich potential. Then, a i and b j can be uniquely determined by the marginal constraint [78]) proved, in the continous measure framework, that a unique KL-projection exists and takes the form γ(x, y) = a(x) ⊗ b(y)γ(x, y) (where a(x) and b(y) are non-negative functions).…”

Section: Numericsmentioning

confidence: 99%