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Random Matrix Theory and Gaussian Noise Thresholding

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This Demonstration shows an application of random matrix theory (RMT) to the problem of signal-from-noise separation in large real-valued symmetric random matrices that takes advantage of the RMT predictions about the nearest-neighbor spacing distribution (NNSD) between the eigenvalues of these matrices.

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Two scenarios are possible:

1. The matrix elements are completely random. In this case, the NNSD is described by the Gaussian orthogonal ensemble (GOE) statistics and its shape can be approximated by the Wigner surmise function.

2. The matrix has a block-like modular structure. In this case, the NNSD shape is described by an exponential function.

If Gaussian noise is added to a structured block-like matrix, the transition from the exponential to the GOE regime can be verified.

In this Demonstration, the controls "matrix size" and "matrix blocks" let you set the size of the matrix and the number of blocks. The control "de-noising threshold " lets you adjust the level of Gaussian noise added to the matrix. If it is set at its maximum (slider completely to the right), the matrix is bare, that is, no noise is added.

By setting the control "matrix blocks" to 1, the matrix is made by a unique block of random elements and its corresponding NNSD always obey the GOE statistics, whatever the setting of the de-noising threshold. If the control "matrix blocks" is set to a value greater than 1, then the matrix has a block-like structure. If in this case the "de-noising threshold " is moved from left to right, the structured signal emerges from the matrix plot and you can observe the transition of the NNSD from the GOE distribution (i.e. with noise added) to the exponential regime (structured signal).

In this Demonstration, the eigenvalue spacings have been rescaled to match the RMT universal distributions (GOE, exponential) according to the density distribution of eigenvalues by implementing a proper unfolding procedure.

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Contributed by: Jessica Alfonsi  (August 2020)
(Padova, Italy)
Open content licensed under CC BY-NC-SA

Snapshots

Details

Snapshot 1: structured block matrix with no Gaussian noise

Snapshot 2: eigenvalue spacing distribution computed from the matrix in Snapshot 1 following an exponential function

Snapshot 3: masked structured block matrix with superimposed Gaussian noise

Snapshot 4: eigenvalue spacing distribution computed from the matrix in Snapshot 3 following a Wigner surmise function (Gaussian orthogonal ensemble distribution)

This work was inspired by Uwe Menzel's introduction to his own authored R package detailed in [1, 2].

References

[1] U. Menzel. "RMThreshold: Signal-Noise Separation in Random Matrices by Using Eigenvalue Spectrum Analysis." (Aug 18, 2020) cran.r-project.org/web/packages/RMThreshold.

[2] U. Menzel. "RMThreshold: A Short Introduction." (Aug 18, 2020) www.matstat.org/content_en/RMT/RMThreshold_Intro.pdf.

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