Nonlinear feature extraction techniques for understanding sensory data with application to speech

Abstract

Speech is one of the most preferred way of human communication. Although it can be represented as a one-dimensional time series, the process of its generation and perception is quite nonlinear, and the attempts to model and understand its structure have ended up with limited success. In this thesis, a general framework of understanding sensory data is provided using theories from nonlinear dynamics. More specifically, a number of useful theorems on the recoverability of the dynamical system behind the observed sensory data are presented. Guided by the theoretical results, several recently proposed nonlinear machine learning techniques are improved and applied to speech data to uncover its structure and solve practical problems such as fundamental frequency estimation and speech recognition. Contributions of this thesis can be summarized as follows: First, a theorem on the recoverability of controllable dynamical systems using a delay embedding map is proved by extending Takens` theorem. It is shown that by imposing some constraints on the maximum and minimum periods of the deterministic dynamical system $\It{M}$, one can approximately reconstruct the product manifold of parameters $\It{N}$and the attractors of underlying dynamical systems $\It{M}$ given only one-dimensional observation. This opens up the possibility of estimating hidden parameters of a sensory signal without having too much domain-specific knowledge on it. The theorem provides some guidance about picking the dimensionality of the delay embedding. Also, a theorem bounding the error of the reconstruction is given. The reconstruction error of the delay embedding map is bounded if we assume a $(D,\delta)$-slow parameter trajectory with $\delta$ small enough, and goes to zero as $\delta \rightarrow 0$. In other words, if the parameter governing the passive dynamical system changes slow enough, then we can reconstruct the product manifold. Second, the method of recovering the controllable dynamical...