This is a simple tool to compute the probability densities of eigenvalue distributions of the information plus noise random matrices.
We consider an information plus noise random matrices as follows: for a given p x d matrix A and a real number \sigma > 0, define a random matrix
where Z is a p x d (real or complex) Ginibre random matrix (i.e. whose entries are i.i.d. and distributed with N(0, sqrt(1/d)) ). If the size N is large enough, eigenvalue distribution of W can be approximated by deterministic probability distribution on positive real line. This tool computes the probability density function of the deterministic eigenvalue distribution.
Our argorithm is based on the papers Operator-valued Semicircular Elements: Solving A Quadratic Matrix Equation with Positivity Constraints (J. William Helton, Reza Rashidi Far and Roland Speicher, 2007, IMRN) and Free Probability Theory: Deterministic Equivalents and Combinatorics(Carlos Vargas Obieta, 2015)
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p=240,d=40, A = rectangular_diag(0,0.1,0.2, ..., 3.9), \sigma = 0.1:
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p=d=64, A = 0, \sigma = 0.1 (Marchenko-Pastur type):
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p=d=64, A = Identity matrix, \sigma = 0.1:
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p=d=64, A = diag(0.2,0.2..., 0.5, 0.5,...) (32 of 0.2 and 32 0.5), v = 0.1:
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p=d=64, A = diag(1,2,3, ..., 64)/80, \sigma = 0.1:
python 3, numpy, scipy, matplotlib. We recommend Anaconda.
$ git clone https://github.com/ThayaFluss/fde-ipn.git$ cd fde-ipn
$ python -m unittest tests/*.py$ cd fde-ipn
$ ipythonimport numpy as np
from matrix_util import *
from demo_cauchy import *
dim = 64
p_dim = dim
scale = 1e-3
sigma = 0.1
min_x = -0.1
max_x = 0.5
a = 5*np.ones(dim)
half = int(dim/2)
for i in range(half):
a[i] = 2
diag_A = a/10
A = rectangular_diag(diag_A,p_dim, dim)
num_shot = 200 ### The number of sample matrices. If we set num_shot=0, only density is plotted.
plot_density(A, sigma,scale, min_x, max_x, \
num_shot=num_shot,jobname="2-5") To plot all figures in this README, try
$ cd fde-ipn
$ python demo_plots.py