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.{{cite journal|last1=Cardoso|first1=Jean-François|last2=Souloumiac|first2=Antoine|title=Blind beamforming for non-Gaussian signals|journal=IEE Proceedings F - Radar and Signal Processing |date=1993|volume=140|issue=6|pages=362–370|doi=10.1049/ip-f-2.1993.0054|citeseerx=10.1.1.8.5684}} 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 $\mathbf{X} = (x_{ij}) \in \mathbb{R}^{m \times n}$ denote an observed data matrix whose $n$ columns correspond to observations of $m$ -variate mixed vectors. It is assumed that $\mathbf{X}$ 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,

{{Center| $\frac{1}{n}\sum_{j=1}^n x_{ij} = 0 \quad \text{and} \quad \frac{1}{n}\mathbf{X}{\mathbf X}^{\prime} = \mathbf{I}_m$ . }}

Applying JADE to $\mathbf{X}$ entails

computing fourth-order cumulants of $\mathbf{X}$ and then
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 $\mathbf{Z} := \mathbf{O}^{-1} \mathbf{X}$ .{{cite journal|last1=Cardoso|first1=Jean-François|title=High-order contrasts for independent component analysis|journal=Neural Computation|date=Jan 1999|volume=11|issue=1|pages=157–192|doi=10.1162/089976699300016863|citeseerx=10.1.1.308.8611}}

References

Category:Computational statistics