Recently, Yann Ollivier developed a nice theory of Ricci curvature for Markov chains. In many ways, this can be seen as a geometric language giving another view on the notion of path coupling, developed at the end of the ‘s by Martin Dyer and co-workers. It has to be noted that this new notion of curvature is very general and does not need the state space where the Markov chain evolves to have any differential structure, as can be expected at first sight. Any state space endowed with a metric suffices.
Let be a Markov kernel on a metric state space . We would like to quantify how long it takes for two different particles evolving according to the Markovian dynamic given by to meet. If the first particle starts at and the second at , the initial distance between them is . At time , what is the average distance between these two particles. For example, if and are two Brownian motions in started from and respectively, there is no reason why and should be closer from each other than and . Indeed, one can even show that whatever the coupling of these two Brownian motions we have : this is roughly speaking because the Euclidean space has no curvature. The situation is quite different if we were instead considering Brownian motions on a sphere: in this case, trajectories tend to coalesce.
1. Wasserstein distance
In the sequel, we will need to use a notion of distance between probability distributions on the metric space . The usual total variation distance defined by
is not adapted to our purpose since the metric structure of the space is not exploited. Instead, in order to take into account the distance of the space and develop a notion of curvature, we use the Wasserstein distance between probability measures. It is defined as
The distance is crucial to this definition: a change of distance implies a change of the class of -Lipschitz functions. Since for any coupling of and , and since the function is -Lipschitz, it follows that . Consequently, for any coupling we have . Taking the infimum over all the couplings leads to the inequality
This is a deep result that on any reasonable space the inequality is in fact an equality. Indeed, Kantorovich duality states that on any Radon space we have
It is interesting to note that under mild conditions on the state space one can always find a coupling that achieves the infimum of (4): this is an easy compactness argument.
2. Notion of Curvature
Denoting by the one step distribution of the Markov chain started from in the sense that , we define the local (Ricci) curvature between and as
The closer to is , the more the trajectories started at tend to meet the trajectories started at .
The interesting case is when the infimum is strictly positive,
In this case we say that the Markov kernel is positively curved on . It should be noted that in many natural spaces it suffices to ensure that for all neighbouring states and to ensure that for any pair . This can be proved thanks to the so called Gluing Lemma. A space without curvature correspond to the case : for example, a symmetric random walk on and a Brownian motion on have both zero curvature. The curvature is a property of both the metric space and the Markov kernel : indeed, different Markov chain on the same metric space have generally different associated curvature. Given a metric space carrying a probability distribution , this is an interesting problem to construct a -invariant Markov chain with the highest possible curvature .
Indeed, the notion of curvature readily generalizes to continuous time Markov processes by taking a limiting case of (5). For example, one can define the curvature of the continuous time Markov process as the largest real number such that for any and we have
for every small enough. The quantity is the distribution of when started from in the sense that .
3. Contraction property
We now show that a positive curvature implies a contraction property. Equation (5) shows that for any . A simple argument shows that one can indeed generalize the situation to any two distributions in the sense that
Proof: For any pair consider a coupling of and such that . Now, choose an optimal coupling of and . This is straightforward to check that is a coupling (in general not optimal) of and so that
Equation (8) is extremely powerful since it immediately shows that
In other words, there is exponential convergence (in the Wasserstein metric) to the invariance distribution at rate . In continuous time, this reads
In other words, the higher the curvature, the faster the convergence to equilibrium.
4. Examples
Let us give examples of positively curved Markov chains.
- Langevin diffusion with convex potential: consider a convex potential that is uniformly elliptic in the sense . The Langevin diffusion has invariant distribution with density proportional to . Given a time step , the Euler discretization of this diffusion reads
where . Given two starting points and , using the same noise to define and it immediately follows that
In other words, the Langevin diffusion is positively curved with curvature (at least) equal to .
- Brownian motion on a sphere: consider a Brownian motion on the unit sphere of . Consider two points on this unit sphere: by symmetry, one can always rotate the coordinates so that that and for some . For the (geodesic) distance is approximated by . One can couple two Brownian motions and , one started at and the other one started at , by the usual symmetry with respect to the plane : in other words, is the reflexion of with respect to . One can check (good exercise!) that the diffusion followed by the -coordinate of a Brownian motion on the unit sphere of is simply given by
With this coupling, for small time , it follows that
where is used as the same source of randomness for and since is the reflexion of . Since it readily follows that
In other words, the curvature of a Brownian motion on the unit sphere of is equal to . Maybe surprisingly, the higher the dimension, the faster the convergence to equilibrium. This is not so unreal if one notices that the Brownian increment satisfies .
- Other examples: see the original text for many other examples.