For convex manifolds, the manifold centroid
is the center of mass in Figure 5.8 and is on the manifold.
It is also the conditional mean
and is an optimal point with respect to a deterministic entropy
measure. It has applications in feature inversion,
image reconstruction and spectral estimation.
Predicting the Mean (centroid) of MCMC-UMS.
The centroid can be approximated by the sample mean
of samples generated using UMS, or
much more efficiently using the ``surrogate density"
approach, which we now explain. Let
, be a PDF with support on
(not limited to the manifold), but sharing four properties with
(a) its mean lies on the manifold, so
(b) it has constant density along the manifold (meaning that
the gradient in a direction aligned with the manifold is zero),
(c) it has maximum possible entropy under the
constraint (5.12), and (d) it has support in all of ,
but itsprobability mass is concentrated
near the manifold. This idea is illustrated in Figure 5.9.
The property that the samples congregate near the manifold
for large can be justified by the law of large numbers
(See Appendix in ). As a result,
the surrogate density converges effectively to the manifold distribution.
Therefore, the mean
of the surrogate density
is a very good approximation to the manifold centroid
at high dimensions.
The property that the surrogate distribution is uniform
along the manifold can be seen once we select the
surrogate density and maximize its entropy.
It is known that the exponential density
has the highest entropy among all densities
for positive-valued with specified mean
Illustration of surrogate density. An arbitrary sample
is decomposed into a component in the column space
of and the orthogonal component . At high dimension,
samples congregate near the manifold where
and are equally distributed along the manifold.
We therefore propose to use (5.13)
as the surrogate density for
by maximizing the entropy of (5.13)
, subject to
The entropy of (5.13) is
where ``p" indicates positive data case.
If we use (5.10) to write
in terms of ,
we can maximize over . The solution must meet the requirement
that the derivatives of the entropy with respect to
are zero, or
This condition forces the distribution to be constant on the manifold.
To see this, first, let be decomposed as (See Figure 5.9),
, where matrix spans the subspace orthogonal
to matrix . Note that changes to vector
will move within the manifold, but not change
its projection onto the columns of ,
so remains on the manifold.
Therefore, a distribution is constant on the manifold
if and only if its derivative w/r to is zero.
It is easily shown that the derivative of
with respect to equals
making (5.15) equivalent to requiring
to be constant on the manifold.
Incidentally, note that maximizing (5.14)
also maximizes the classical maximum entropy measure
which is used in classical
spectral estimation [34,35] and
image reconstruction [36,37].
There are two ways to solve (5.15), one requiring
a valid starting point in the manifold, and one not
requiring a starting point.