Joint Approximation Diagonalization of Eigen-matrices explained

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.

Algorithm

Let

X = (x _{ij} ) \in R ^{m} 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 x m dimensional identity matrix, that is,Applying JADE to X entails

1. computing fourth-order cumulants of

X and then

1. optimizing a contrast function to obtain a

m x m rotation matrix O to estimate the source components given by the rows of the m x n dimensional matrix Z : = O ^{-1} X .^[2]

Notes and References

1. Cardoso. Jean-François. Souloumiac. Antoine. Blind beamforming for non-Gaussian signals. IEE Proceedings F - Radar and Signal Processing . 1993. 140. 6. 362–370. 10.1049/ip-f-2.1993.0054. 10.1.1.8.5684.
2. Cardoso. Jean-François. High-order contrasts for independent component analysis. Neural Computation. Jan 1999. 11. 1. 157–192. 10.1162/089976699300016863. 10.1.1.308.8611.