Aane Transformations of the Speech Space

The papers Speaker Normalization of static and dynamic vowel spectral features by Zahorian and Jagharghi describe an algorithm for transforming the space of speech sounds so as to improve the accuracy of classiication. Classiication was accomplished by both back-propagation neural nets and by a Bayesian Maximum Likelihood method on the model of each vowel class being speciied by a gaussian distribution. The transformation was an aane transformation obtained by choosing ideaìtarget' points for each cluster in a second space and minimising the mean square distance of the points in the speech space from the appropriate target. The speech space itself was a space of cepstral coeecients obtained from a Discrete Cosine Transform. These ndings are remarkable, indeed almost unbelievable. The reason is that both the maximum likelihood classiication on the gaussian model, and the Neural Net classiier are essentially aane invariant. In the case where the transformation is invertible, this is clearly the case. When the transformation has non-trivial kernel, it may happen that the classiication gets worse, but it cannot get better. A back-propagation neural net in eeect classiies by dividing the space into regions by means of hyperplanes. The gaussian model does so by means of quadratic forms, with quadratic discrimination hypersurfaces. Projecting a hyperplane by any non-zero aane map which is onto the target space will usually give another hyperplane in the target space, and if the second separates points, so will the rst. Conversely, if there is a solution in the target space, it can be pulled back to a solution in the domain space. It is not hard to show that similar considerations apply to the case where we use quadratic hypersurfaces. In this paper, we attempt to account for the results of Zahorian and Jagharghi by investigating vowel data. We describe a simple projection algorithm which may be applied to high dimensional data to give a view on a computer screen of the data and of transformations of it.