Kernel-independent component analysis

Kernel independent component analysis (Kernel ICA) is an efficient algorithm for independent component analysis which estimates source components by optimizing a generalized variance contrast function, which is based on representations in a reproducing kernel Hilbert space. Those contrast functions use the notion of mutual information as a measure of statistical independence.

Main idea

Kernel ICA is based on the idea that correlations between two random variables can be represented in a reproducing kernel Hilbert space (RKHS), denoted by ${\mathcal {F}}$, associated with a feature map $L_{x}:{\mathcal {F}}\mapsto \mathbb {R}$ defined for a fixed $x\in \mathbb {R}$. The ${\mathcal {F}}$-correlation between two random variables $X$ and $Y$ is defined

$\rho _{\mathcal {F}}(X,Y)=\max _{f,g\in {\mathcal {F}}}{\text{corr}}(\langle \ L_{X},f\rangle ,\langle \ L_{Y},g\rangle )$

where the functions $f,g:\mathbb {R} \to \mathbb {R}$ range over ${\mathcal {F}}$ and

${\text{corr}}(\langle \ L_{X},f\rangle ,\langle \ L_{Y},g\rangle ):={\frac {{\text{cov}}(f(X),g(Y))}{{\text{var}}(f(X))^{1/2}{\text{var}}(g(Y))^{1/2}}}$

for fixed $f,g\in {\mathcal {F}}$. Note that the reproducing property implies that $f(x)=\langle \ L_{x},f\rangle$ for fixed $x\in \mathbb {R}$ and $f\in {\mathcal {F}}$. It follows then that the ${\mathcal {F}}$-correlation between two independent random variables is zero.

This notion of ${\mathcal {F}}$-correlations is used for defining contrast functions that are optimized in the Kernel ICA algorithm. Specifically, if $\mathbf {X} :=(x_{ij})\in \mathbb {R} ^{n\times m}$ is a prewhitened data matrix, that is, the sample mean of each column is zero and the sample covariance of the rows is the $m\times m$ dimensional identity matrix, Kernel ICA estimates a $m\times m$ dimensional orthogonal matrix $\mathbf {A}$ so as to minimize finite-sample ${\mathcal {F}}$-correlations between the columns of $\mathbf {S} :=\mathbf {X} \mathbf {A} ^{\prime }$.