CNL

Cauchy Noise Loss

This is a tool for optimization of two random matrix models; compound Wishart model and signal-plus-noise (information-plus-noise) model.

Description

The compound Wishart model is $W = Z^TAZ$, where $Z$ is a (real or complex) $p \times d$ Ginibre random matrix (i.e. whose entries are i.i.d. and distributed with $N(0, \sqrt(1/d)))$, and A is $p \times p$ deterministic self-adjoint matrix.

We consider a signal-plus-noise random matrices as follows: for a given $p \times d$ matrix $A$ and a real number $\sigma > 0$, define a random matrix $$ Y = A + \sigma Z,\ W = Y^*Y $$

If $d$ is large enough and $p= O(d)$, eigenvalue distribution of $Y^* Y$ can be approximated by deterministic probability distribution on positive real line.

Our algorithm is based on the paper "Cauchy noise loss for stochastic optimization of random matrix models via free deterministic equivalents (https://arxiv.org/abs/1804.03154)".

Requirement

python 3.6, cython, numpy, scipy, tqdm, matplotlib. We recommend to use the platform Anaconda.

Installation

$ git clone https://github.com/ThayaFluss/cnl.git

Setup

$ bash setup.bash
cd src
python -m unittest tests/*.py

Usage

cd src

For the probabilistic singular value decomposition based on Cauchy noise loss;

 X #numpy.array of shape [p,d]
 from psvd import *
 U,D,V, sigma = psvd_cnl(X)

For the rank estimation of $p \times d$ matrix $X$;

 X #numpy.array of shape [p,d]
 from psvd import *
 rank, a, sigma = rank_estimation(X) #estimated rank and parameters  a, sigma.

For example; (https://github.com/ThayaFluss/cnl/blob/master/demo_rank_estimation.py)

 p_dim = 100
 dim = 50
 min_singular = 0.3
 true_rank = 30
 import numpy as np
 a_true = np.random.uniform(min_singular, 1, dim)
 for i in range(dim-true_rank):
   a_true[i] = 0  
 from matrix_util import *
 D = rectangular_diag(a_true, p_dim=p_dim, dim=dim)
 from random_matrices import *
 U = haar_unitary(p_dim)
 V = haar_unitary(dim)
 A_true = U @ D @ V ; #random rotation
 X = signal_plus_noise(A_true, sigma=0.1) ### sample matrix
 from psvd import *
 rank, a, sigma = rank_estimation(X) ### estimated rank and parameters
 print(rank, true_rank) ### compare with the true_rank !

Validation

To validate algorithms in the same way as the numerical experiments in the paper;

git checkout arXiv1804
cd src
$ python validate_train_sc.py
$ python validate_train_cw.py

License

MIT