To visualize the sampling distribution, we
can generate random samples on the manifold for a fixed
feature value, and plot the pair
for two indexes on a plane.
For a simple example, we let the feature be the two statistics
and . Figure 4.1 shows a simulation for and .
The initial feature value was created by
drawing a random
from a standard
normal distribution, and computing .
With fixed, we generated random samples of on the manifold.
On the left, we see random samples of plotted on
the plane. On the right, we see a histogram of
. Interestingly, for a 2-dimensional manifold, ,
the samples will fall on an elliptical curve.
One can see that with increasing manifold dimension,
the marginal distributions approach Gaussian,
even though the sampling is uniform on the manifold.
This can be understood since it is well known that
normalizing a vector of independent Gaussian
random variables by its 2-norm generates points
uniformly distributed on a hyper-sphere .
Since the normalization constant converges
to a constant at high dimension, it follows that the
distribution of is approximately Gaussian.
Asymptotic (large ) behavior
From top, manifold sampling results for ,
, and . Left: random samples of , .
Right, histogram of .