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Given a data matrix, $M \in \mathbb{R}^{n \times p}$, I am interested in methods quantifying the amount of structure in present in $M$.

I've found a few approaches, but I would like to learn more about what is available and what can be done given new developments in random matrix theory. I understand that the phrase "quantifying the amount of structure" is quite vague, any references that can clear that up would be appreciated.

Currently, I know about the following methods:

-- Compare the leading singular value of $M$ against the leading singular value of a shuffled version of $M$ (cf. http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.0030160).

Here, one determines whether or not $M$ has structure by comparing the leading singular value of $M$, $\sigma_{max}(M)$, with the leading singular value of a matrix, $\tilde{M}$, obtained by permuting the entries of $M$. If $| \sigma_{max}(M) - \sigma_{max}(\tilde{M}) | / \sigma_{max}(M) < 0.15$ then $M$ is said to contain no structure. Otherwise, the leading rank-1 approximation to $M$ is subtracted off and the process is repeated until no structure remains.

-- Compare the leading singular value of $M$ against the Tracy-Widom distribution (cf. http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aos/1009210544 and http://www.plosgenetics.org/article/info:doi/10.1371/journal.pgen.0020190 )

This is identical to the previous test with the exception that we test against the Tracy-Widom distribution rather than the singular values of a shuffled version of $M$. I suppose this test is more rigorous but will only tell whether or not the entries of $M$ are Gaussian ( though I get the impression that newer results improve on this, eg http://arxiv.org/pdf/0906.0510v10.pdf ).

-- Check how well the correlation eigenvalues of $M$ agree with the semicircular law (cf. http://arxiv.org/pdf/cond-mat/0108023v1.pdf )

Here, the authors study the structure of several time series by observing how many eigenvalues lie within Wigner's semicircle. It would be nice if the number of significant principle components one gets from the methods above is equal to the number of eigenvalues which lie outside the semicircle. I have no idea if this is true.

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Since your data matrix is rectangular, in your last proposed test you presumably want to compare the eigenvalue distribution of $MM^*$ to the Marchenko-Pastur distribution, not the semicircle law. –  Mark Meckes Aug 23 '12 at 10:46
    
Thanks. I will look into that now. –  rcompton Aug 23 '12 at 22:20
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