Lazy random walks and optimal transport on graphs

Léonard, Christian

doi:10.1214/15-aop1012

Cited by 28 publications

(44 citation statements)

References 38 publications

(44 reference statements)

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“…When weights represent costs, our approach of Section VI compromizing between minimization and robustness can be further compared to Optimal Mass Transport (OMT) over graphs [30], where only cost matters, and entropically regularized OMT-schemes [31], [32]. In discrete OMT, however, the cost function is supposed to be given, although computing it is typically an intractable problem for large networks.…”

Section: Discussionmentioning

confidence: 99%

Robust Transport Over Networks

Chen

Georgiou

Pavon

et al. 2017

IEEE Trans. Automat. Contr.

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We consider transportation over a strongly connected, directed graph. The scheduling amounts to selecting transition probabilities for a discrete-time Markov evolution which is designed to be consistent with initial and final marginal constraints on mass transport. We address the situation where initially the mass is concentrated on certain nodes and needs to be transported in a certain time period to another set of nodes, possibly disjoint from the first. The random evolution is selected to be closest to a prior measure on paths in the relative entropy sense–such a construction is known as a Schrödinger bridge between the two given marginals. It may be viewed as an atypical stochastic control problem where the control consists in suitably modifying the prior transition mechanism. The prior can be chosen to incorporate constraints and costs for traversing specific edges of the graph, but it can also be selected to allocate equal probability to all paths of equal length connecting any two nodes (i.e., a uniform distribution on paths). This latter choice for prior transitions relies on the so-called Ruelle-Bowen random walker and gives rise to scheduling that tends to utilize all paths as uniformly as the topology allows. Thus, this Ruelle-Bowen law (𝔐RB) taken as prior, leads to a transportation plan that tends to lessen congestion and ensures a level of robustness. We also show that the distribution 𝔐RB on paths, which attains the maximum entropy rate for the random walker given by the topological entropy, can itself be obtained as the time-homogeneous solution of a maximum entropy problem for measures on paths (also a Schrödinger bridge problem, albeit with prior that is not a probability measure). Finally we show that the paradigm of Schrödinger bridges as a mechanism for scheduling transport on networks can be adapted to graphs that are not strongly connected, as well as to weighted graphs. In the latter case, our approach may be used to design a transportation plan which effectively compromises between robustness and other criteria such as cost. Indeed, we explicitly provide a robust transportation plan which assigns maximum probability to minimum cost paths and therefore compares favourably with Optimal Mass Transportation strategies.

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Section: Discussionmentioning

confidence: 99%

Robust Transport Over Networks

Chen

Georgiou

Pavon

et al. 2017

IEEE Trans. Automat. Contr.

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“…An important consequence of this convergence result is that the limit µ t := lim k→∞ P k t in the Definition 2.3 of the displacement interpolation is effective under mild hypotheses, see [Léo12,Léoa].…”

Section: Informal Statement 211 (See [Léo12 Léoa])mentioning

confidence: 94%

“…Most of the present material comes from the papers [Léo12,Léoa,Léob]. Very little about literature is proposed; more is given in the previously cited papers.…”

Section: Introductionmentioning

confidence: 99%

Some recent developments in functional inequalities

et al. 2014

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“…A survey of the main results of this theory is found in [Leo14]. A construction of entropic interpolations and a discussion of the cases where they can be described as mixtures of binomials is found in [Leo13b]. Another paper, see [Leo13a], addresses the question of the convexity of entropy along such interpolations.…”

Section: Existence Of W 1+ -Geodesicsmentioning

confidence: 99%

$W_{1,+}$-interpolation of probability measures on graphs

Hillion

2014

Electron. J. Probab.

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We generalize an equation introduced by Benamou and Brenier in [BB00] and characterizing Wasserstein Wp-geodesics for p > 1, from the continuous setting of probability distributions on a Riemannian manifold to the discrete setting of probability distributions on a general graph. Given an initial and a final distributions (f0(x))x∈G, (f1(x))x∈G, we prove the existence of a curve (ft(k)) t∈[0,1],k∈Z satisfying this Benamou-Brenier equation. We also show that such a curve can be described as a mixture of binomial distributions with respect to a coupling that is solution of a certain optimization problem

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Lazy random walks and optimal transport on graphs

Cited by 28 publications

References 38 publications

Robust Transport Over Networks

Robust Transport Over Networks

Some recent developments in functional inequalities

$W_{1,+}$-interpolation of probability measures on graphs

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