Complex Principle Kurtosis Analysis

Independent component analysis (ICA) is a fundamental problem in the field of signal processing, and numerous algorithms have been developed to address this issue. The core principle of these algorithms is to find a transformation matrix that maximizes the non-Gaussianity of the separated signals. Most algorithms typically assume that the source signals are mutually independent (orthogonal to each other), thereby imposing an orthogonal constraint on the transformation matrix. However, this assumption is not always valid in practical scenarios, where the orthogonal constraint can lead to inaccurate results. Recently, tensor-based algorithms have attracted much attention due to their ability to reduce computational complexity and enhance separation performance. In these algorithms, ICA is reformulated as an eigenpair problem of a statistical tensor. Importantly, the eigenpairs of a tensor are not inherently orthogonal, making tensor-based algorithms more suitable for nonorthogonal cases. Despite this advantage, finding exact solutions to the tensor's eigenpair problem remains a challenging task. In this paper, we introduce a non-zero volume constraint and a Riemannian gradient-based algorithm to solve the tensor's eigenpair problem. The proposed algorithm can find exact solutions under nonorthogonal conditions, making it more effective for separating nonorthogonal sources. Additionally, existing tensor-based algorithms typically rely on third-order statistics and are limited to real-valued data. To overcome this limitation, we extend tensor-based algorithms to the complex domain by constructing a fourth-order statistical tensor. Experiments conducted on both synthetic and real-world datasets demonstrate the effectiveness of the proposed algorithm.