A generalization of the Baum algorithm to functions on non-linear manifolds

The well-known Baum-Eagon (1967) inequality provides an effective iterative scheme for homogeneous polynomials with positive coefficients over a domain of probability values Δ. The Baum-Eagon inequality was extended to rational functions over Δ by Gopalakrishnan et. al. (see IEEE Trans. Inform. Theory, Jan. 1991) and a variant of this extended inequality was used by Merialdo (see Proc. ICASSP-88, 1988, and IEEE Trans. Acoust., Speech, Signal Processing, April 1994) for the maximum mutual information training of a connected digit recognizer. However, in many applications (e.g. corrective training) we are interested in maximizing an objective function over a domain D that is different from Δ and may be defined by non-linear constraints. We show how to extend the basic inequality of Gopalakrishnan to (not necessary rational) functions that are defined on general manifolds. We describe an effective iterative scheme that is based on this inequality and its application to estimation problems via minimum information discrimination