### Method 2 (not requiring starting point)

To solve (5.15), define as the element-wise inverse of . Then, (5.15) can be written , which means that must be fully in the column space of , or that

 (5.17)

for some vector . Therefore, to find the mean of the surrogate density, we solve for the free variable such that

 (5.18)

where

 (5.19)

We can solve this by driving the square error to zero:

 (5.20)

We can easily find derivative of with respect to :

 (5.21)

where is the diagonal matrix with diagonal elements

 (5.22)

We have found that if we use the negative-definite Hessian approximation

 (5.23)

the resulting Newton-Raphson algorithm has excellent convergence properties when starting with . Let be the value of at the solution to (5.18). The algorithm to find can be summarized as follows.
1. Set iteration counter .
2. To initialize, let where is the vector of ones.
3. . Initially, will be the vector of ones.
4. Compute derivative and Hessian according to (5.21),(5.22) and (5.23), then update :

5. Increment and go to step 3.
The algorithm is implemnted by software/lam_solve.m with dbound=0. Although this method corresponds to classical methods, it is based on a novel sampling argument and can be extend to other manifolds as we will see.

Baggenstoss 2017-05-19