Ricci curvature of Markov chains on metric spaces

By Yann Ollivier. 2007

909 citations recorded by the source corpus.

Abstract We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scaling, the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein–Uhlenbeck process. Moreover this generalization is consistent with the Bakry–Emery Ricci curvature for Brownian motion with a drift on a Riemannian manifold. Positive Ricci curvature is shown to imply a spectral gap, a Levy–Gromov–like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. The bounds obtained are sharp in a variety of examples.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/1311becf342434a949ce5fba51d9b024e052850f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Generalized couplings and ergodic rates for SPDEs and other Markov models (2018, background): “In the terms of [35], conditions 3 and 4 of Proposition 2.”
The Markov chain Monte Carlo revolution (2008, methodology): “The very useful path coupling variant in [11] and [35] is developed into a marvelous theory of Ricci curvature for Markov chains by [73].”
A spectral notion of Gromov–Wasserstein distance and related methods (2011, methodology): “By exploiting the connections of Ricci curvature with mass transportation concepts, the work of Lott and Villani [49], Sturm [50] and Ollivier [51] has expanded considerably the understanding of how curvature may be defined on metric measure spaces and how it may play a role in guaranteeing compactn”
Exponential ergodicity for Markov processes with random switching (2013, background): “ation (1.3) is not much of a restriction because we do not assume that ρ(i)>0. The best constant in this inequality is called the Wasserstein curvaturein [26, 27] and the coarse Ricci curvature in [29, 30], since it is heavily related to the geometry of the underlying space as illustrated in [34]”
Li–Yau inequality on finite graphs via non-linear curvature dimension conditions (2014, neutral): “[17, 18, 19, 21]).”
Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure (2012, methodology): “This implies that RCD(K,∞) spaces have Ricci curvature bounded from below by K according to the W1-contractivity property taken as a definition in Ollivier [26] and Joulin [22].”
Curvature and transport inequalities for Markov chains in discrete spaces (2015, methodology): “Note that T1 cannot hold in the full generality of the setting of [37], since it implies Gaussian concentration, which does not hold for some examples with positive curvature, more details and examples can be found in [37].”
Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries (2011, background): “[43] Y. Ollivier, Ricci curvature of Markov chains on metric spaces J. Funct.”
Bakry–Émery curvature and diameter bounds on graphs (2016, background): “However, for many discrete curvature notions there is already a BonnetMyers-type result established (for sectional curvature on planar graphs, see [6, 10, 14, 22], and for Ollivier-Ricci-curvature, see [20], and for Forman’s discrete Ricci curvature, see [8]).”
The Geometric Foundations of Hamiltonian Monte Carlo (2014, methodology): “Initial applications of metric space methods [41,55], for example, have shown promise [35], although many technical obstacles, such as the limitations of geodesic completeness, remain.”
Quantitative contraction rates for Markov chains on general state spaces (2018, background): “ rates. Contractivity with respect to the L1 Wasserstein distance based on the Euclidean distance in Rd is sometimes interpreted as non-negative Ricci curvature of the Markov chain w.r.t. this metric [41, 29, 37]. This is a strong condition that is often not satisﬁed in applications. However, in man”
On the spectrum of hypergraphs (2017, methodology): “The spectrum of normalized Laplacian operator on a connected hypergraph is also bounded by the Olliviers Ricci curvature of the hypergraph.”
Ricci Curvature of Finite Markov Chains via Convexity of the Entropy (2011, background): “In the setting of graphs Ollivier’s Ricci curvature has been further studied in the recent preprints [6, 22, 24].”
Ricci curvature of graphs (2011, methodology): “We will use similar notations as in [20].”
Discrete Curvature and Abelian Groups (2015, background): “fact considered and discussed in [33] in 1999, but seems to have largely been neglected ever since. For additional and more recent approaches to discrete Ricci curvature and related inequalities, see [6, 16, 19, 27, 30, 32, 34]. The fact that one can conclude from positive (or negative) curvature, a”
Cutoff for non-negatively curved Markov chains (2021, background): “More recently, Ollivier [53] put forward a different definition of curvature based on optimal transport, which applies to arbitrary metric spaces.”