Absolute continuity and summability of transport densities: simpler proofs and new estimates

Santambrogio, Filippo

doi:10.1007/s00526-009-0231-8

Cited by 53 publications

(74 citation statements)

References 18 publications

(44 reference statements)

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“…Several avenues for future research will address remaining theoretical and practical challenges suggested by our work. On the theoretical side, an analog of our analysis may apply to the Beckmann model of transport over R n [1,32], showing how quadratic regularization affects flows in the continuum.…”

Section: Discussionmentioning

confidence: 96%

“…These algorithms are primarily of theoretical interest but do employ interior point-style methods, possibly implying a systematic choice of flows in the case of multiple optima.In the continuum, the theory of optimal transport [42] classifies problems structured similarly to minimum-cost flow as the 1-Wasserstein distance or Beckmann problem [1]. See [33] for analysis and [42,17,32] for theoretical discussion. Even over general spaces, solutions of the 1-Wasserstein problem generally are nonunique and include some degenerate optima [42, §2.4.6].…”

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confidence: 99%

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Quadratically Regularized Optimal Transport on Graphs

Essid¹,

Solomon²

2018

SIAM J. Sci. Comput.

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Optimal transportation provides a means of lifting distances between points on a geometric domain to distances between signals over the domain, expressed as probability distributions. On a graph, transportation problems can be used to express challenging tasks involving matching supply to demand with minimal shipment expense; in discrete language, these become minimumcost network flow problems. Regularization typically is needed to ensure uniqueness for the linear ground distance case and to improve optimization convergence; state-of-the-art techniques employ entropic regularization on the transportation matrix. In this paper, we explore a quadratic alternative to entropic regularization for transport over a graph. We theoretically analyze the behavior of quadratically-regularized graph transport, characterizing how regularization affects the structure of flows in the regime of small but nonzero regularization. We further exploit elegant second-order structure in the dual of this problem to derive an easily-implemented Newton-type optimization algorithm.Transport over graphs has a rich history and has been considered in mathematics, operations research, computer science, and many other disciplines. At the broadest level, this problem is a slight generalization of the linear assignment problem and can be solved using several classical techniques; see e.g. [6,4] for discussion.Without regularization, the particular problem we study would be an instance of minimum-cost flow without edge capacities.[30] discusses classical algorithms for this linear programming problem, such as the cycle canceling [20], network simplex [28], and Ford-Fulkerson algorithms [15]. Classical methods such as these often are not accompanied with systematic "tie-breaking" strategies in case the network flow is non-unique, indicating the potential application of a strictly convex variation of the problem such as ours.The theoretical computer science community recently has reconsidered this class of problems from an optimization perspective. In the undirected, capacity-free case, [36] proposes a preconditioner for approximate minimum-cost flow problems that achieves nearly-linear runtimes.[10] considers the more general case of minimum-cost flow in directed graphs with unit capacities, extending a framework proposed in [26] for approaching graph-based problems using the interior point method. These algorithms are primarily of theoretical interest but do employ interior point-style methods, possibly implying a systematic choice of flows in the case of multiple optima.In the continuum, the theory of optimal transport [42] classifies problems structured similarly to minimum-cost flow as the 1-Wasserstein distance or Beckmann problem [1]. See [33] for analysis and [42,17,32] for theoretical discussion. Even over general spaces, solutions of the 1-Wasserstein problem generally are nonunique and include some degenerate optima [42, §2.4.6]. Numerical algorithms for these problems include [39], which uses finite element methods for a vector field ver...

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Section: Discussionmentioning

confidence: 96%

mentioning

confidence: 99%

Quadratically Regularized Optimal Transport on Graphs

Essid¹,

Solomon²

2018

SIAM J. Sci. Comput.

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“…Indeed, (2) first arose as the "Beckmann problem" in network flow [Beckmann 1952]. See [Santambrogio 2013] for further analysis of this problem and its connection to optimal transportation, and see [Villani 2003, §1.2.3], [Feldman and McCann 2002], and [Santambrogio 2009] for a broader discussion.…”

Section: Optimal Transportationmentioning

confidence: 99%

Earth mover's distances on discrete surfaces

et al. 2014

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We introduce a novel method for computing the earth mover's distance (EMD) between probability distributions on a discrete surface. Rather than using a large linear program with a quadratic number of variables, we apply the theory of optimal transportation and pass to a dual differential formulation with linear scaling. After discretization using finite elements (FEM) and development of an accompanying optimization method, we apply our new EMD to problems in graphics and geometry processing. In particular, we uncover a class of smooth distances on a surface transitioning from a purely spectral distance to the geodesic distance between points; these distances also can be extended to the volume inside and outside the surface. A number of additional applications of our machinery to geometry problems in graphics are presented.

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“…The L p regularity of the transport density σ is proved successively by many authors (see, for instance, [7,8,9,11,18]). In particular, we have the following…”

Section: Introductionmentioning

confidence: 96%

Lack of regularity of the transport density in the Monge problem

Dweik

2018

Journal de Mathématiques Pures et Appliquées

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Abstract. In this paper, we provide a family of counter-examples to the regularity of the transport density in the classical Monge-Kantorovich problem. We prove that the W 1,p regularity of the source and target measures f ± does not imply that the transport density σ is W 1,p , that the BV regularity of f ± does not imply that σ is BV and that f ± ∈ C ∞ does not imply that σ is W 1,p , for large p.

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Absolute continuity and summability of transport densities: simpler proofs and new estimates

Cited by 53 publications

References 18 publications

Quadratically Regularized Optimal Transport on Graphs

Quadratically Regularized Optimal Transport on Graphs

Earth mover's distances on discrete surfaces

Lack of regularity of the transport density in the Monge problem

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