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.¹ 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.

Let ${\displaystyle \mathbf {X} =(x_{ij})\in \mathbb {R} ^{m\times n}}$ denote an observed data matrix whose ${\displaystyle n}$ columns correspond to observations of ${\displaystyle m}$-variate mixed vectors. It is assumed that ${\displaystyle \mathbf {X} }$ is prewhitened, that is, its rows have a sample mean equaling zero and a sample covariance is the ${\displaystyle m\times m}$ dimensional identity matrix, that is,

${\displaystyle {\frac {1}{n}}\sum _{j=1}^{n}x_{ij}=0\quad {\text{and}}\quad {\frac {1}{n}}\mathbf {X} {\mathbf {X} }^{\prime }=\mathbf {I} _{m}}$.

Applying JADE to ${\displaystyle \mathbf {X} }$ entails

1. computing fourth-order cumulants of ${\displaystyle \mathbf {X} }$ and then
2. optimizing a contrast function to obtain a ${\displaystyle m\times m}$ rotation matrix ${\displaystyle O}$

to estimate the source components given by the rows of the ${\displaystyle m\times n}$ dimensional matrix ${\displaystyle \mathbf {Z} :=\mathbf {O} ^{-1}\mathbf {X} }$.²