Research On The Theory Of Maximum Nongaussianity Estimation In Independent Component Analysis

In the recent years, Independent Component Analysis (ICA) has become a hot topic in signal processing and neural networks. For its outstanding performance in blind identification and feature extraction (or representation), ICA has rapidly growing applications in various fields, e.g., telecommunication systems, speech processing (blind extraction), image enhancement, and biomedical signal processing. Different from the approaches based on the principle of nonlinear decorrelation, e.g., Maximum Likehood Estimation (MLE), Minimum Mutual Information (MMI), maximization of nongaussianity is a classical one-unit approach to find the linear transformation for observed signals such that the output has a local maximum nongaussianty, and then discover one of the latent indenpendnt components. Compared with the multi-unit approaches, maximum nongaussinity estimation is superiror in simple theoretical foundation and flexible implementation, and thereof the popular fast fixed-point iteration (FastICA) based on negentropy approximation has become one of the most popular algorithms for ICA.The dissertation is dedicatd to the theory of maximum nongaussianity estimation in ICA, and some problems are emphasized, e.g., the uniqueness of estimation, the convergence of FastICA, the constraint of independence and the nonlinear activation function, the algorithms for multiple components and the order of independnent components. The main contributions of this paper include:1. The uniqueness of maximum nongaussianity The property of uniqueness is crucial for the theory of maximum nongaussianity estimation and the guarantee for the availability of the algorithms for ICA and the applications. For maximum nongaussianity estimation, uniqueness means that there is a one-to-one correspondence between local maxima of nongaussianity and independent sources. Different from the former works, e.g., the heuristic analysis, some concepts and theorems of constrained optimization are employed, e.g., linear feasible direction (LFD), and then the proof is first provided for the uniqueness, i.e., maximum nongaussianity is the sufficient and necessary condition to independent source recovery. Moreover, based on the conclusion of the uniqueness of maximum nongaussianity, the paper also proposes an alternative but straightforward approach to the proof for the well known"one-bit-matching"conjecture presented in multi-unit approaches.2. The global convergence of FastICA Based on the deduction of FastICA, the global convergence analysis is given. In a two-channel system, the exhaustive equilibria are evaluated first, and then the relationship between initial demixing vectors and the equilibria is disclosed. The result shows that there are no spurious solutions in FastICA when the basic assumptions in ICA model are satisfied. And that the conclusion can also be generalized to multi-channel system. In addition, some practical considerations, e.g.,...