Growth transform of a sum of rational functions and its application in estimating HMM parameters

Gopalakrishnan et al [1] 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 [5, 6, 9]. 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 [8]. We will show that the growth transform of a sum of rational functions can be obtained by computing term-wise gradients and term-wise function values, as opposed to forming rst a single rational function and then applying the result in [1]. 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 nd the appropriate transform constant C.