Joint Approximation Diagonalization of Eigen-matrices

Short description: Independent component analysis algorithm

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

[math]\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 }[/math].

Applying JADE to [math]\displaystyle{ \mathbf{X} }[/math] entails

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

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