Random Matrix Theory in a nutshell and applications Manuela Girotti - - PowerPoint PPT Presentation

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Apr 21, 2023 586 likes •724 views

Random Matrix Theory in a nutshell and applications Manuela Girotti IFT 6085, February 27th, 2020 1 / 14 Random Matrix Theory Consider a matrix A : a 11 a 12 a 13 . . . . . . a 1 N a 21 a 22 a 23 . . . a 2 N a 31 .

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Random Matrix Theory in a nutshell and 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_n x_2 x_3 ...

Their (probable) positions will depend upon the probability distribution of the entries of the matrix in a non-trivial way. There are different statistical quantities that one may study (-discrete- spectral density, 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 dimension n = 20, 50, 100.

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Figure: Histograms of the eigenvalues of GUE matrices as the size of the matrix increases.

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Figure: Histograms of the eigenvalues of GUE matrices as the size of the matrix increases.

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Figure: Histogram of the eigenvalues of Wishart matrices as the size of the matrix increases (here c = p/n).

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Applications

Nuclear Physics (distribution of the energy levels of highly excited states 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 matrix

1 n

∇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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