## Predicting the Mean (centroid) of MCMC-UMS.

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.

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

 (5.12)

(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 [21]). 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 [33].

 (5.13)

We therefore propose to use (5.13) as the surrogate density for , by maximizing the entropy of (5.13) over , subject to     The entropy of (5.13) is

 (5.14)

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

 (5.15)

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

 (5.16)

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.

Subsections
Baggenstoss 2017-05-19