Joint Approximation Diagonalization of Eigen-matrices

Joint Approximation Diagonalization of Eigen-matrices (JADE) is an algorithm for independent component analysis that separates observed mixed signals into latent source signals by exploiting fourth order moments. The fourth order moments are a measure of non-Gaussianity, which is used as a proxy for defining independence between the source signals. The motivation for this measure is that Gaussian distributions possess zero excess kurtosis, and with non-Gaussianity being a canonical assumption of ICA, JADE seeks an orthogonal rotation of the observed mixed vectors to estimate source vectors which possess high values of excess kurtosis.

Algorithm
Let $$\mathbf{X} = (x_{ij}) \in \mathbb{R}^{m \times n}$$ denote an observed data matrix whose $$n$$ columns correspond to observations of $$m$$-variate mixed vectors. It is assumed that $$\mathbf{X}$$ is prewhitened, that is, its rows have a sample mean equaling zero and a sample covariance is the $$m \times m$$ dimensional identity matrix, that is, $ \frac{1}{n}\sum_{j=1}^n x_{ij} = 0 \quad \text{and} \quad \frac{1}{n}\mathbf{X}{\mathbf X}^{\prime} = \mathbf{I}_m $. Applying JADE to $$\mathbf{X}$$ entails to estimate the source components given by the rows of the $$m \times n$$ dimensional matrix $$\mathbf{Z} := \mathbf{O}^{-1} \mathbf{X}$$.
 * 1) computing fourth-order cumulants of $$\mathbf{X}$$ and then
 * 2) optimizing a contrast function to obtain a $$m \times m$$ rotation matrix $$O$$