Here's what I'm trying to do:

Imagine a probability distribution over $\mathbf{R}^2$, $P(x,y)$. I can approximate $P(x,y)$ with set of $N$ points $\{(x,y)_i\}$ drawn from $P$. By approximate, I mean that for a test function $\phi$, I'll have $$\int\phi dP \simeq \frac{1}{N}\sum_i\phi(x_i,y_i)$$

Now imagine that the random variables x and y are strongly correlated, not necessarily in the sense of Pearson correlation, but in the sense that $y \simeq f(x)$. The support of the probability distribution will be very narrow, almost 1-dimensional. This means that I can get a good approximation with a smaller value of N... ( the variance of $\int\phi dP - \sum_i\phi(x_i,y_i)$ will be much lower than it would, if the support was wider )

This is all well and I'm happy.

Now assume instead the opposite case, $x$ and $y$ are completely independent. This implies that $P$ is separable, $P(x,y) = P_x(x)P_y(y)$

In practice, I am dealing with two 1-dimensional distribution, but the support of $P$ can be very wide. Now, if I approximate $P$ with a cloud of $N$ points, and if my test function doesn't depend on one of the variable, I will still get very good estimates with a small N, because I am oversampling the $P_x(x)$ across all $y$. However, if $\phi$ is localized in the plane, I will get a much noisier result than I would, if I sampled $P_x$ and $P_y$ independently.

We're getting to my problem!

I have a distribution $P$ over $\mathbf{R}^n$, and I know that the variables $(x,y,z,\ldots)$ are either quite correlated or quite independent from each other. I'd like to find a map $M: \mathbf{R}^n \rightarrow \mathbf{R}^n $ so that $N$ random samples from $P \circ M$ will give a good approximation.

Maybe another way to do it would be to systematically sample from every marginal distribution, and write $P(x,y) = P_x(x)P_y(y) + Q(x,y)$... or I could do a PCA first, and separate based on those components... but notice how I need to make assumptions here (a multivariate normal relationship for example), whereas in the dependent case, the random sample works beautifully and adapts to any kind of dependency? I feel like I'm missing something more generic here.

For a practical example, pick a time and date in the past year and consider the system ( earth, moon, seconds since last hour) in heliocentric coordinates. I picked second since last hour instead of say, date, so that the position of the earth and moon doesn't depend explicitly on time.

The vector is $(e_x,e_y,e_z,m_x,m_y,m_z,t)$. The marginal distribution of $t$ is uniform, and $(e_x,e_y,e_z,m_x,m_y,m_z)$ is largely independent of $t$ and lives around a 2-dimensional manifold (one parametrization being given by two angles). All in all, our random sample will sample around a 3-dimensional manifold. That's not so good. We could see that time is separable and instead sample the 2-d system (earth,moon) and the 1-d system (time). We could do even better and parametrize the system with two angles, and see that those are separable... This example could be described with three 1-dimensional distributions, but how? This parametrization is not obvious.