Starting point

As we mentioned, we need to start MCMC-UMS with a valid $ {\bf x}$. Inversion of the dimension-reducing transformation $ {\bf z}={\bf A}^\prime {\bf x}$ using pseudo-inverse

$\displaystyle {\bf x}_p = {\bf A}({\bf A}^\prime {\bf A})^{-1} {\bf z}^*,$ (5.11)

produces a solution for $ {\bf x}$ that meets the linear constraint $ {\bf A}^\prime {\bf x}= {\bf z}^*$, but may have negative-valued elements. A good initial starting point can be obtained from any linear-programming solver. Specifically, we find the solution to the maximization of $ q=\sum_{i=1}^N x_i $ subject to $ {\bf x}\in {\cal M}({\bf z}^*).$ It makes no difference if we are maximizing or minimizing since $ q$ is fixed anyway by the linear constraints. Any linear programming (LP) algorithm such as OCTAVE glpk.m or MATLAB linprog.m can output a ``solution" which is a valid point. See software/module_A_chisq_synth for additional details. The best starting point, however, is the manifold centroid, discused in Section 5.3.2.

Baggenstoss 2017-05-19