Weak convergence of measures and Regularized optimal transport

Question

Let W_p be the Wasserstein metric on the space of probability measures on $\mathcal{X}$, a compact subset of $\mathbb{R}^n$. I know that $\mu_k \rightarrow \mu$ (under weak convergence of measures) iff $W_p(\mu_k, \mu) \rightarrow 0$. Now instead of working with $W_p$, I am using the entropic regularized version of $W_p$, call it $W_{p, \lambda}$ as introduced in https://papers.nips.cc/paper/2013/file/af21d0c97db2e27e13572cbf59eb343d-Paper.pdf. Due to some pathological properties of $W_{p, \lambda}$, let us assume that the sequence $\mu_k$ is not constant and also the number of samples drawn from our measures are arbitrarily large. Then my question is: Is it still true that $\mu_k \rightarrow \mu$ iff $W_{p, \lambda}(\mu_k, \mu) \rightarrow 0$

Answer 1

So I’ve found the answer. [This article by Feydey et al][1] proves that for a compact metric space $X$, and a Lipschitz cost $c$. Then the Sinkhorn Divergence does metrize the weak convergence, i.e $\tilde W_{p,\epsilon}(\mu_n,\mu)\to 0 \iff \mu_n \rightharpoonup \mu$.

Just note that the Sinkhorn divergence is an unbiased version of the regularized cost, where:

$$ \tilde W_{p,\epsilon}(\alpha, \beta)^p = 2W_{p,\epsilon}(\alpha,\beta) -W_{p,\epsilon}(\alpha,\alpha)-W_{p,\epsilon}(\beta,\beta) $$ [1]: https://hal.archives-ouvertes.fr/hal-01898858/document