Nonlinear Dimensionality Reduction Methods for Use with Automatic Speech Recognition

For nearly a century, researchers have investigated and used mathematical techniques for reducing the dimensionality of vector valued data used to characterize categorical data with the goal of preserving “information” or discriminability of the different categories in the reduced dimensionality data. The most established techniques are Principal Components Analysis (PCA) and Linear Discriminant Analysis (LDA) (Jolliffe, 1986; Wang & Paliwal, 2003). Both PCA and LDA are based on linear, i.e. matrix multiplication, transformations. For the case of PCA, the transformation is based on minimizing mean square error between original data vectors and data vectors that can be estimated from the reduced dimensionality data vectors. For the case of LDA, the transformation is based on minimizing a ratio of “between class variance” to “within class variance” with the goal of reducing data variation in the same class and increasing the separation between classes. There are newer versions of these methods such as Heteroscedastic Discriminant Analysis (HDA) (Kumar & Andreou, 1998; Saon et al., 2000). However, in all cases certain assumptions are made about the statistical properties of the original data (such as multivariate Gaussian); even more fundamentally, the transformations are restricted to be linear. In this chapter, a class of nonlinear transformations is presented both from a theoretical and experimental point of view. Theoretically, the nonlinear methods have the potential to be more “efficient” than linear methods, that is, give better representations with fewer dimensions. In addition, some examples are shown from experiments with Automatic Speech Recognition (ASR) where the nonlinear methods in fact perform better, resulting in higher ASR accuracy than obtained with either the original speech features, or linearly reduced feature sets. Two nonlinear transformation methods, along with several variations, are presented. In one of these methods, referred to as nonlinear PCA (NLPCA), the goal of the nonlinear transformation is to minimize the mean square error between features estimated from reduced dimensionality features and original features. Thus this method is patterned after PCA. In the second method, referred to as nonlinear LDA (NLDA), the goal of the nonlinear transformation is to maximize discriminability of categories of data. Thus the method is patterned after LDA. In all cases, the dimensionality reduction is accomplished with a Neural Network (NN), which internally encodes data with a reduced number of dimensions. The differences in the methods depend on error criteria used to train the network, the architecture of the network, and the extent to which the reduced dimensions are “hidden” in the neural network.