  This note considers softmax parameter estimation when little/no labeled
training data is available, but a priori information about the relative
geometry of class label log-odds boundaries is available. It is shown that
`data-free' softmax model synthesis corresponds to solving a linear system of
parameter equations, wherein desired dominant class log-odds boundaries are
encoded via convex polytopes that decompose the input feature space. When
solvable, the linear equations yield closed-form softmax parameter solution
families using class boundary polytope specifications only. This allows softmax
parameter learning to be implemented without expensive brute force data
sampling and numerical optimization. The linear equations can also be adapted
to constrained maximum likelihood estimation in data-sparse settings. Since
solutions may also fail to exist for the linear parameter equations derived
from certain polytope specifications, it is thus also shown that there exist
probabilistic classification problems over m convexly separable classes for
which the log-odds boundaries cannot be learned using an m-class softmax model.
