Cardoso's Independent Component Analysis (ICA)

Cardoso's Independent Component Analysis (ICA) represents a classical blind source separation technique primarily used to recover original independent source signals from mixed observations. The Joint Approximate Diagonalization of Eigenmatrices (JADE) method, proposed by Cardoso, serves as an efficient ICA implementation particularly suitable for solving multidimensional signal separation problems.

The core principle of the JADE method relies on higher-order statistics, where it computes eigenmatrices of signals and performs joint diagonalization to identify the optimal separation matrix. Unlike traditional ICA approaches, JADE eliminates the need for distributional assumptions by directly utilizing fourth-order cumulants (kurtosis) to optimize the objective function. This implementation approach enables more stable estimation of independent components, especially for non-Gaussian signals. In code implementation, this typically involves calculating the cumulant tensor and performing eigenvalue decomposition.

Key algorithmic steps in JADE include: initially computing higher-order statistical matrices (typically fourth-order cumulant matrices) of observed signals, then performing joint diagonalization on these matrices to ultimately obtain the separation matrix. The joint diagonalization process essentially seeks a transformation matrix that makes multiple eigenmatrices approach diagonal structure as closely as possible, thereby ensuring component independence. Since JADE employs algebraic optimization rather than iterative computation, it generally outperforms gradient-descent-based ICA methods in computational efficiency. A typical implementation would involve whitening preprocessing followed by joint diagonalization using Jacobi rotations.

The advantages of the JADE method lie in its mathematical rigor and computational efficiency, making it applicable across multiple domains including speech signal processing, EEG signal analysis, and image feature extraction. However, its performance depends heavily on higher-order statistical characteristics of signals, meaning that for mixed signals with strong Gaussian noise, additional preprocessing methods may be required to enhance separation effectiveness. Common implementation considerations include proper data normalization and dimension reduction techniques.