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,
1 n ∑ j = 1 n x i j = 0 and 1 n X X ′ = I m {\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 X {\displaystyle \mathbf {X} } entails
- computing fourth-order cumulants of X {\displaystyle \mathbf {X} } and then
- 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} } .2
References
Cardoso, Jean-François; Souloumiac, Antoine (1993). "Blind beamforming for non-Gaussian signals". IEE Proceedings F - Radar and Signal Processing. 140 (6): 362–370. CiteSeerX 10.1.1.8.5684. doi:10.1049/ip-f-2.1993.0054. /wiki/CiteSeerX_(identifier) ↩
Cardoso, Jean-François (Jan 1999). "High-order contrasts for independent component analysis". Neural Computation. 11 (1): 157–192. CiteSeerX 10.1.1.308.8611. doi:10.1162/089976699300016863. /wiki/CiteSeerX_(identifier) ↩