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 $𝐗=(x_{ij})\in {ℝ}^{m\times n}$ denote an observed data matrix whose $n$ columns correspond to observations of $m$-variate mixed vectors. It is assumed that $𝐗$ 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,

$\frac{1}{n}\sum _{j=1}^n{x}_{ij}=0\text{and}\frac{1}{n}𝐗{𝐗}^{\prime }={𝐈}_m$.

Applying JADE to $𝐗$ entails

1. computing fourth-order cumulants of $𝐗$ and then
2. optimizing a contrast function 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 $𝐙:={𝐎}^{-1}𝐗$.