Joint Approximation Diagonalization of Eigen-matrices

Joint Approximation Diagonalisation of Eigenmatrices (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 proxy measure for non-Gaussianity, which is used for defining independence between the source signals.

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 prewhitenend, 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

1. computing the fourth cumulants of $\mathbf {X}$ and then
2. optimizing an orthogonal contrast to obtain a $m\times m$ rotation matrix $O$

to estimate the source components given by the rows of the $m\times n$ dimensional matrix $\mathbf {Z} :=\mathbf {O} ^{-1}\mathbf {X}$.