Integrating Gaussian mixtures into deep neural networks: Softmax layer with hidden variables

In the hybrid approach, neural network output directly serves as hidden Markov model (HMM) state posterior probability estimates. In contrast to this, in the tandem approach neural network output is used as input features to improve classic Gaussian mixture model (GMM) based emission probability estimates. This paper shows that GMM can be easily integrated into the deep neural network framework. By exploiting its equivalence with the log-linear mixture model (LMM), GMM can be transformed to a large softmax layer followed by a summation pooling layer. Theoretical and experimental results indicate that the jointly trained and optimally chosen GMM and bottleneck tandem features cannot perform worse than a hybrid model. Thus, the question “hybrid vs. tandem” simplifies to optimizing the output layer of a neural network. Speech recognition experiments are carried out on a broadcast news and conversations task using up to 12 feed-forward hidden layers with sigmoid and rectified linear unit activation functions. The evaluation of the LMM layer shows recognition gains over the classic softmax output.