Dimension of the speech space

Abstract

Tattersal et al. have described an attempt to use the Kohonen algorithm for locating a hypothetical two dimensional speech space in a space of filter bank values. The existence of this space is moot and its dimension is more moot, and the Kohonen algorithm itself does not yield any information on the intrinsic dimension of the set to which it converges. There is therefore some interest in trying to decide by other means whether or not the speech space does have an intrinsic dimension of two. This paper falls into two main sections. In the first, we define a statistic for estimating the intrinsic dimension of a finite set of points on the assumption that they lie on a smoothly embedded manifold, when, of course, the dimension is an integer. We test the method on finite sets drawn from known manifolds and show that it is robust. We also apply it to the Lorenz attractor, which is a well known set of nonintegral dimension. Finally we apply it to speech data of the same type as that used by Tattersal et al. We conclude that the speech space is not discernibly a low dimensional manifold at all, and that a more plausible hypothesis is that the space is an open subset of the enclosing space. In the second section, we construct a measure of the extent to which the surface that the Kohonen algorithm fits to the speech space is buckled or crinkled related to the mean absolute curvature. The intention is to test (a) the hypothesis that the points of the speech space constitute a muralium, a twomanifold with noise, and (b) the hypothesis that the Kohonen process will find the muralium. We conclude that it is indeed possible to approximate the speech space with a low dimensional manifold, but that it has dimension greater than two.