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Canonical Correlation Analysis #18

@loiseaujc

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@loiseaujc

For component analysis, fsml currently provides only pca (and eof), i.e. the diagonalization of the correlation matrix $C_{xx}$. Another very common tool is Canonical Correlation Analysis. Given two random vector variables $x$ and $y$, it essentially amounts to simultaneously diagonalize $C_{xx}$, $C_{yy}$ and $C_{xy}$ and does so by solving the following eigenvalue problems

$$ \lambda p = C_{xx}^{-\dfrac12} C_{xy} C_{yy}^{-1} C_{yx} C_{xx}^{-\dfrac12} $$

and

$$ \lambda q = C_{yy}^{-\dfrac12} C_{yx} C_{xx}^{-1} C_{xy} C_{yy}^{-\dfrac12}. $$

Then $p_i^\top C_xx p_j = \delta_{ij}$, $q_i^\top C_yy q_j = \delta_{ij}$ and $p_i^\top C_{xy} q_j = \lambda_i \delta_{ij}$. Note that if $x$ and $y$ are Gaussian random vectors, the $\lambda$'s are related to the mutual information between the two random variables.

Would the addition of such a component analysis technique be of interest to you?

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