5th International Conference on Spoken Language Processing
Gopalakrishnan et al described a method called "growth transform" to optimize rational functions over a domain, which has been found useful to train discriminatively Hidden Markov Models(HMM) in speech recognition. A sum of rational functions is encountered when the contributions from other HMM states are weighted in estimating Gaussian parameters of a state, and the weights are optimized using cross- validation. We will show that the growth transform of a sum of rational function can be obtained by computing term-wise gradients and term-wise function values, as opposed to forming first a single rational function and then applying the result in [Gopal91]. This is computationally advantageous when the objective function consists of many rational terms and the dimensionality of the domain is high. We also propose a gradient directed search algorithm to find the appropriate transform constant C.
Bibliographic reference. Luo, Xiaoqiang (1998): "Growth transform of a sum of rational functions and its application in estimating HMM parameters", In ICSLP-1998, paper 0364.