Uniform sampling on a Riemannian manifold via tangent space and exponential map

Question

Given a Riemannian manifold $(\mathcal{M}, \{g_x\}_{x \in \mathcal{M}})$ and a fixed point $x \in \mathcal{M}$, does the following procedure yield uniform samples from $\{y \in \mathcal{M} : d_\mathcal{M}(x, y) \le 1 \}$?

1. Sample uniformly from $\{u \in \mathcal{T}_x \mathcal{M} : \lVert u \rVert_x \le 1\}$ (say we know how to do that)
2. $y = \exp_x(u)$

It seems to me that by change of variables the density over $y$ will be constant only if the determinant of the Jacobian of $\exp_x(\cdot)$ is constant. But I'm not sure if this argument is correct mainly because I don't know if the pushforward measure of the metric-induced measure on the set from step 1 via the exponential map is the same as the volume form induced on $\mathcal{M}$ by $g_x$.

(I'm a novice in both Riemannian geometry and measure theory, so I'm sorry if this is trivially true or false, or it doesn't make sense.)

Answer 1

As Nate Eldredge pointed out in his comment, your two-step procedure will not in general simulate (a random element of $\mathcal M$ with) the uniform density $f=1/|B_x|$ on $B_x:=\{y\in\mathcal M\colon d_{\mathcal M}(y,x)\le1\}$, where $|B_x|$ is the volume of $B_x$. However, this "two-step" density, say $h$, can be used to simulate $f$, using the so-called rejection sampling method, as follows.

Suppose that $c\in[1,\infty)$ is such that $$f(y)\le c\,h(y)\tag{1}$$ for all $y\in B_x$. Let $Y$ be a random element of $B_x$ with density $h$, and let $U$ be a random variable uniformly distributed in the interval $[0,1]$ and independent of $Y$. Then the random pair $(Y,Uc\,h(Y))$ will be uniformly distributed on the subgraph $\{(y,t)\colon y\in B_x,0\le t\le c\,h(y)\}$ of the function $c\,h$ on $B_x$.

Now simulate a value $y$ of $Y$ using your two-step procedure, and then independently simulate a value $u$ of $U$.

Next, if $u\,c\,h(y)>f(y)$, discard the simulated value $y$ of $Y$ (as well as the simulated value $u$ of $U$). Otherwise, i.e. if $u\,c\,h(y)\le f(y)$, accept the simulated value $y$ of $Y$ as the simulated value $x$ of a random element $X$ of $B_x$. Then the random pair $(X,Uc\,h(X))$ will be uniformly distributed on the subgraph $\{(y,t)\colon y\in B_x,0\le t\le f(y)\}$ of the function $f$ on $B_x$ and hence $X$ with have the desired density $f$.

The expected "waste" fraction due to the rejection of some realizations of $Y$ will then be $\int_{B_x}(h(y)-f(y)/c)\,dy=1-1/c$. So, if $h$ is not far from $f$, then we can take the bounding factor $c$ in (1) to be somewhat close to $1$, and then the expected "waste" fraction $1-1/c$ will be somewhat close to $0$.