  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.
