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The Marcenko-Pastur law is about asymptotic distributions of eigenvalues. It starts from a simple null model (iid zero-mean Gaussian entries) and derives a distribution for the spectrum. In PCA, this allows a data analyst to select the number of principal components using a p-value cutoff.

Is there a similar law that governs the distribution of correlations in CCA? I want to select the dimension of the latent space(s) using the same strategy as above.

This question is very similar to mine, but it's not answered and it's a little less general. (Asking for a test about any relationship at all is like asking "Is the latent space is zero dimensional" whereas I want to select the dimension). It's also about kernel CCA, and I am content with just plain CCA.

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    $\begingroup$ +1. I don't know a theoretical answer, but in practice it is usually easy to obtain this distribution via simulation. Generate two matrices of the same size as your X and Y with iid Gaussian entries and compute CCA. One can repeat this several times to get a sense of variability. One can also simply shuffle the rows of Y, to avoid using the Gaussianity assumption. (Shuffling rows of Y destroys all correlations between X and Y, while keeping the covariance of Y unchanged.) $\endgroup$ – amoeba Sep 17 '18 at 15:38

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