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

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

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

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