Random Matrix Theory in a nutshell and applications · Random Matrix Theory in a nutshell and...
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Random Matrix Theory in a nutshell and applications Manuela Girotti IFT 6085, February 27th, 2020 1 / 14
Transcript of Random Matrix Theory in a nutshell and applications · Random Matrix Theory in a nutshell and...
Random Matrix Theory in a nutshelland applications
Manuela Girotti
IFT 6085, February 27th, 2020
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Random Matrix Theory
Consider a matrix A:
A =
a11 a12 a13 . . . . . . a1N
a21 a22 a23 . . . a2N
a31 . . ..........
aM1 aN2 . . . aMN
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where the entries are random numbers:
A =
∗ ∗ ∗ . . . . . . ∗∗ ∗ ∗ . . . ∗∗ . . ..........∗ ∗ . . . ∗
where ∗ =
(Gaussian distribution, e.g.)
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A few questions:
What about the eigenvalues?
x_1 x_nx_2 x_3 ...
Their (probable) positions will depend upon the probabilitydistribution of the entries of the matrix in a non-trivial way. There aredifferent statistical quantities that one may study (-discrete- spectraldensity, gap probability, spacing, etc.)
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What if we consider BIG matrices, possibly of ∞ dimension(appropriately rescaled)?
Figure: Realization of the eigenvalues of a GUE matrix of dimensionn = 20, 50, 100.
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Figure: Histograms of the eigenvalues of GUE matrices as the size of the matrixincreases.
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Figure: Histograms of the eigenvalues of GUE matrices as the size of the matrixincreases.
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Figure: Histogram of the eigenvalues of Wishart matrices as the size of the matrixincreases (here c = p/n).
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Applications
Nuclear Physics (distribution of the energy levels of highly excitedstates of heavy nuclei, say uranium 92U)
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Applications
Wireless communications
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Applications
Finance (stock markets, investment strategies)
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Applications
Very helpful in ML!
Figure: The histogram of the eigenvalues of the gradient covariance matrix1n
∑∇Li∇LT
i for a Resnet-32 with (left) and without (right) BN after 9k
training steps. (from Ghorbani et al., 2019)
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Thanks for your attention!
“Unfortunately, no one can be told what the Matrix is.You have to see it for yourself.”
(Morpheus, “The Matrix” movie)
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