We present a systematic framework for the Nesterov's accelerated gradient flows in the spaces of probabilities embedded with information metrics. Here two metrics are considered, including both the Fisher-Rao metric and the Wasserstein-2 metric. For the Wasserstein-2 metric case, we prove the convergence properties of the accelerated gradient flows, and introduce their formulations in Gaussian families. Furthermore, we propose a practical discrete-time algorithm in particle implementations with an adaptive restart technique. We formulate a novel bandwidth selection method, which learns the Wasserstein-2 gradient direction from Brownian-motion samples. Experimental results including Bayesian inference show the strength of the current method compared with the state-of-the-art.
I acknowledge reading the rebuttal of the authors. Thank you for your clarifications and explanation. My point was this paper would make a good submission to ICLR if it was better motivated presented and explained to a wider audience. Unfortunately in its current form it can only reach a limited audience.
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Summary of the paper: 
The paper proposes accelerated information flows under Wasserstein or Fisher rao metric . i.e a method for solving minimization of probability functional , where the probability space is either endowed with the Wasserstein or the Fisher distance.
Gradient descent methods in euclidian spaces can be accelerated using a type of momentum , this paper extends this to gradient flows, using similar formalism of Hamiltonian that appeals to the dynamic of a the particles and the velocity (or momentum, $H(x,p)=\frac{1}{2}||p||^2+ \mathcal{E}(x)$).  Writing down the lagrangian one obtains two co-evolving PDE one of the dynamic of the density and one for the potential . The PDEs are specified for both Fisher Rao, and Wasserstein distance. The hamiltonian for example for the Wasserstein distance $H(\rho_t,\Phi_t)=\frac{1}{2}\int||\nabla_x \Phi_t||^2+ \mathcal{E}(\rho_t)$.  and PDEs amounts a continuity equation for the density  evolving with drift $\nabla \Phi_t$, the evolution of the momentum $\Phi_t$ is also given by a PDE. 
Proposition 2 of the paper gives the particles differential equation corresponding to the system of PDEs. For the energy being the KL divergence an explicit expression is given , this expression remains difficult in practice since it needs the knowledge of the density $\rho_t$. Authors propose in the application section to use gaussian approximation , or using a kernel density estimators. The Bandwidth of the kernel is then choosen using a heuristic proposed in the paper.  
The paper then focuses on deriving expression for flows when the densities are centered gaussians, and this amounts to an ODE on the covariance , the ODE is discretized in Appendix D.2 to lead to  computational method. Then convergence of the flow is analyzed for the wasserstein accelerated flows, under "\beta- convextiy" in the wasserstein sense of the functional. 
Some experiments of the particle based method are shown on synthetic experiments and in bayesian logistic regression. 
Review: 
Contribution/ Clarity:
The main contribution of the paper is in deriving the accelerated gradient flow for the wasserstein distance this was also addressed in a recent paper [Taghevia and Mettha 2019]. 
The technical contribution is interesting but given that this field of flows in probability space is still not very well spread in the ML community, I wonder if ICLR is the best fit for this type of work.  I support good theoretical work, but I think the authors could have done a better job in exposing the ideas how they extend form euclidean space, to manifolds, to probability spaces gradually. Simple derivations of euclidean space Hamiltonian will help the reader that is not exposed to such literature. I think the paper will benefit from a less technical writing in introducing the ideas coming from euclidean space and in conveying the intuitions. 
Comments: 
- In the proof of Proposition 2 you give the expression of evolution of $dV_t$ by conservation of the momentum. Could you please elaborate more how you obtain this expression, and where you proved the conservation of momentum?
- In term of damping if ones uses the Wasserstein Fisher Rao flows  , one obtain also accelerartion , maybe you can comment on that ? since you analyze both flows , would be interesting to discuss the relation to Global convergence of neuron birth-death dynamics, that shows that an acceleration is obtained via WFR flows, since it will introduce a damping as well. 
- since MCMC and BM method lead to similar result what is the advantage of the wasserstein accelerated flow? one could also implement also an accelerated langevin dynamic