Note that (6.8) is equivalent to the statement
that the vector
is in the column space of ,
or that there exists a free variable such that
As an aside, note that condition (6.8) also assures that (6.5)
will have zero derivative along the manifold, which is one of
the assumptions of the surrogate density.
The method to find the centroid is to find the
vector such that
is equation (6.3) applied element-wise.
This is essentially the same as for the
positive- case (5.18) , except that the non-linear relationship
The algorithm of Section
5.3.2 to find
driving (5.20) to zero can be used if
the diagonal elements of
are given by
be the value of
at the solution to (6.10).
The modified algorithm to find
The above algorithm is implemented by software/lam_solve.m with dbound=1.
- Set iteration counter .
- To initialize, let
will be the
vector of zeros.
using (6.3) element-wise.
- Compute derivative and Hessian according to (5.21) and (5.23),
and (6.11), then update :
- Increment and go to step 3.