A Curved Brunn--Minkowski Inequality on the Discrete Hypercube, Or: What Is the Ricci Curvature of the Discrete Hypercube?

Ollivier, Yann; Villani, Cédric

doi:10.1137/11085966x

Cited by 45 publications

(41 citation statements)

References 17 publications

(28 reference statements)

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“…There are several possible ways to extend the notion of Ricci curvature from Riemannian geometry to the general setting of metric spaces and, in particular, graphs [Cha96, Oll09, Pet11, BJL12]. We use the approach of Ollivier [Oll09,OV12]. In the following discussion G is a finite multigraph with a probability measure m x on its vertex set V = V (G) assigned to each vertex x.…”

Section: Discrete Curvature On Graphsmentioning

confidence: 99%

On Coset Leader Graphs of Structured Linear Codes

Iceland

Samorodnitsky

2019

Discrete Comput Geom

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We suggest a new approach to obtain bounds on locally correctable and some locally testable binary linear codes, by arguing that these codes (or their subcodes) have coset leader graphs with high discrete Ricci curvature.The bounds we obtain for locally correctable codes are worse than the best known bounds obtained using quantum information theory, but are better than those obtained using other methods, such as the "usual" information theory. (We remark that our methods are completely elementary.)The bounds we obtain for a family of locally testable codes improve the best known bounds.

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Section: Discrete Curvature On Graphsmentioning

confidence: 99%

On Coset Leader Graphs of Structured Linear Codes

Iceland

Samorodnitsky

2019

Discrete Comput Geom

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“…There is an extensive literature about Olivier-Ricci curvature on discrete graphs (see for instance, [8], [11], [19], [28], [30], [33], [43], [44], [45] and [46]). and for a function z : X × X → R, its m-divergence div m z : X → R is defined as…”

Section: Introductionmentioning

confidence: 99%

The heat flow on metric random walk spaces

Mazón

Solera

Toledo

2020

Journal of Mathematical Analysis and Applications

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In this paper we study the Heat Flow on Metric Random Walk Spaces, which unifies into a broad framework the heat flow on locally finite weighted connected graphs, the heat flow determined by finite Markov chains and some nonlocal evolution problems. We give different characterizations of the ergodicity and we also prove that a metric random walk space with positive Ollivier-Ricci curvature is ergodic. Furthermore, we prove a Cheeger inequality and, as a consequence, we show that a Poincaré inequality holds if and only if an isoperimetrical inequality holds. We also study the Bakry-Émery curvature-dimension condition and its relation with functional inequalities like the Poincaré inequality and the transport-information inequalities. µ * m(A) := X m x (A)dµ(x), for all Borel sets A ⊂ X,

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“…(i) Ollivier [20] introduces a discrete Ricci curvature via L 1 -Wasserstein metric. Many inequalities on graphs are shown under this setting; see, e.g., [12,13,21]. (ii) Lin-Yau et al [17,18] also define a Ricci curvature lower bound by heat semi-groups and Bakery-Emery Γ 2 operators.…”

Section: Introductionmentioning

confidence: 99%

Ricci curvature for parametric statistics via optimal transport

Montúfar

2020

Info. Geo.

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We elaborate the notion of a Ricci curvature lower bound for parametrized statistical models. Following the seminal ideas of Lott-Strum-Villani, we define this notion based on the geodesic convexity of the Kullback-Leibler divergence in a Wasserstein statistical manifold, that is, a manifold of probability distributions endowed with a Wasserstein metric tensor structure. Within these definitions, the Ricci curvature is related to both, information geometry and Wasserstein geometry. These definitions allow us to formulate bounds on the convergence rate of Wasserstein gradient flows and information functional inequalities in parameter space. We discuss examples of Ricci curvature lower bounds and convergence rates in exponential family models.

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A Curved Brunn--Minkowski Inequality on the Discrete Hypercube, Or: What Is the Ricci Curvature of the Discrete Hypercube?

Cited by 45 publications

References 17 publications

On Coset Leader Graphs of Structured Linear Codes

On Coset Leader Graphs of Structured Linear Codes

The heat flow on metric random walk spaces

Ricci curvature for parametric statistics via optimal transport

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