If we start with an arbitrary $n$-dimensional probability distribution, $\mu$, I am interested in the projection, with respect to the 2-Wasserstein distance, onto the set of product distributions $$Q = \{ \nu: \nu(x) = \prod_{i=1}^n \nu_i(x_i), \int_\mathbb{R} \nu_i(x_i) dx_i = 1, \text{ for } x\in \mathbb{R}^n\}$$
i.e. $Q$ is the set of distributions with its distribution on each component independent of all others.
For any given measure $\mu$, one way to get a sort of corresponding measure in $Q$ is to take the product of all the marginals of $\mu$ (basically ignoring any dependence), $\prod_{i=1}^n \mu(x_i)$. But I would specifically like to get a projection wrt $W_2$ distance,
$$ \text{Proj}_Q(\mu) = \arg\min_{q \in Q} W_2(q, \mu)^2 $$
More specifically, it might be helpful to specify the form of $\mu$: a simple example could be that $\mu$ is a $n$-dimensional Gaussian, and we want the projection onto $n$ independent 1-d Gaussians $q_i(x_i)$. Or, more generally, $\mu$ is an $n$-dimensional exponential family, and we want the projection onto $n$ independent 1-d exponential families.
Any pointers are appreciated.