multivariate Gaussian approximation in total variation distance I'm wondering if there's any general technique that gives the total variation distance between a distribution on $\mathbb{R}^n$ and $N(0, I_n)$.
My understanding is that Stein's method gives only Wasserstein distance in higher dimension because the characterization of multivariate Gaussian is a second-order differential equation (while it is a first-order differential equation in one-dimensional case) so more regularity is required on test functions and thus it yields a weaker distance. And I understand that it is possible to improve Wasserstein distance to total variation distance if the distribution is log-concave.
What is the usual way to handle the total variation distance to multivariate Gaussian? I'm primarily interested in approximating $N(0,I_n)$ but the approximating distribution is not necessarily log-concave. Perhaps there's some easy way for this special case? Or is there any impossibility result?
 A: Stein's method doesn't give total variation approximation in one dimension, either, without some kind of additional assumptions. This has nothing to do with Stein's method; for an impossibility result, any discrete distribution has maximal (1 or 2 depending on your normalization convention) total variation distance to any continuous (e.g. Gaussian) distribution.  But of course you can approximate any distribution by a discrete distribution, in Wasserstein distance for example.
A: I am not sure the following helps.
If $f$ is a prob.distribution on $\mathbb{R}^n$ having the same covariance matrix as $N(0, I_n)$, then \begin{eqnarray*}D(f\|N(0, I_n))&=& H(N(0, I_n))-H(f)\\\ &=& \frac{1}{2}\log((2\pi e)^n)-H(f)\end{eqnarray*}
where $D(\cdot \| \cdot)$ denotes the KL divergence $H$ denotes the Shannon entropy.
Now, by Pinsker's inequality, \begin{eqnarray*}\|f-N(0, I_n)\|_1 &\le& \sqrt{2 D(f\|N(0, I_n))}\\\ &=& \sqrt{\log((2\pi e)^n)-2 H(f)}\end{eqnarray*}
So the conclusion is that if the entropy $f$ is closer to that of Gaussian, $f$ will be closer to $N(0, I_n)$ in total variation. But it works only for those $f$ which also has covariance matrix $I_n$.
