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rmt

Random Matrix Theory: eigenvalue distributions and spectral statistics.

The Core Idea

When you have a large random matrix, its eigenvalues follow predictable distributions. This is surprising: randomness at the element level produces order at the spectral level.

Key Distributions

Distribution	Matrix Type	Density
marchenko_pastur_density	Wishart (X^T X)	Bounded support
wigner_semicircle_density	Symmetric random	Semicircle
Tracy–Widom (edge; not implemented here)	Largest eigenvalue	Skewed

Quick Start

use rmt::{marchenko_pastur_density, wigner_semicircle_density, sample_wishart};
use ndarray::Array2;

// Marchenko-Pastur: eigenvalue density of X^T X / n
let ratio = 0.5;  // p/n (dimensions / samples)
let density = marchenko_pastur_density(1.5, ratio, 1.0);

// Wigner semicircle: eigenvalue density of symmetric matrix
let density = wigner_semicircle_density(0.5, 1.0);

// Sample a Wishart matrix
let (n, p) = (100, 50);
let wishart = sample_wishart(n, p);

Why RMT for ML?

• Covariance matrices: Sample covariance eigenvalues follow Marchenko-Pastur
• Neural networks: Weight matrix spectra reveal training dynamics
• PCA: Distinguish signal from noise eigenvalues
• Regularization: Set shrinkage based on spectral distribution

The Marchenko-Pastur Law

For a matrix X (n samples, p features), the eigenvalues of X^T X / n cluster in [lambda_-, lambda_+] where:

lambda_+/- = sigma^2 (1 +/- sqrt(p/n))^2

Density: rho(lambda) = (1/(2*pi*sigma^2)) * sqrt((lambda_+ - lambda)(lambda - lambda_-)) / (gamma*lambda)

When p/n -> 0, this converges to a point mass at sigma^2 (classical regime). When p/n > 0, eigenvalues spread (high-dimensional regime).

The Wigner Semicircle

For a symmetric matrix with i.i.d. entries, eigenvalues follow:

rho(lambda) = (1/(2*pi*sigma^2)) * sqrt(4*sigma^2 - lambda^2)  for |lambda| <= 2*sigma

What Can Go Wrong

1. Finite size effects: MP/semicircle are asymptotic. Small n deviates.
2. Not centered: MP assumes zero-mean data. Center your features.
3. Correlated features: MP assumes independence. Correlated data has different spectrum.
4. Ratio out of range: MP needs p/n in (0, infinity). Tracy-Widom for edge.
5. Numerical eigendecomposition: For large matrices, use iterative methods.

References

• Marchenko & Pastur (1967). “Distribution of eigenvalues for some sets of random matrices”
• Wigner (1955). “Characteristic vectors of bordered matrices with infinite dimensions”
• Johnstone (2001). “On the distribution of the largest eigenvalue in PCA”

Functions

effective_dimension

Estimate effective dimensionality of an embedding matrix using the Marchenko-Pastur law.

empirical_spectral_density

Empirical spectral density via histogram.

level_spacing_ratios

Level spacing ratio for eigenvalue sequence.

marchenko_pastur_density

Marchenko-Pastur density at point lambda.

marchenko_pastur_support

Marchenko-Pastur support bounds [lambda_-, lambda_+].

mean_spacing_ratio

Mean level spacing ratio.

sample_goe

Sample a GOE (Gaussian Orthogonal Ensemble) matrix.

sample_goe_with

Sample a GOE (Gaussian Orthogonal Ensemble) matrix using the provided RNG for reproducibility.

sample_wishart

Sample a Wishart matrix: W = X^T X where X is n x p Gaussian.

sample_wishart_with

Sample a Wishart matrix W = X^T X where X is n x p Gaussian, using the provided RNG for reproducibility.

stieltjes_transform

Stieltjes transform: m(z) = (1/n) sum 1/(lambda_i - z)

wigner_semicircle_density

Wigner semicircle density at point lambda.