Cholesky Decomposition

Cholesky decomposition (or Cholesky factorization) is a method used to decompose a symmetric, positive-definite matrix into the product of a lower triangular matrix and its transpose.

$A=L{L}^T$

• $A$ is symmetric and positive-definite
• $L$ is lower triangular

I used this to sample gaussian numbers given a Covariance Matrix.

from sklearn.gaussian_process.kernels import RBF
from scipy.linalg import cholesky
 
T = 100
dim = 2
length_scale = 10.0  # Controls smoothness
sigma = 0.1          # Noise level
 
# Define temporal covariance matrix
time_points = np.arange(T).reshape(-1, 1)
kernel = RBF(length_scale=length_scale)
K = kernel(time_points) * sigma**2
 
# Sample correlated noise for each dimension
L = cholesky(K + 1e-6 * np.eye(T))
epsilon = np.dot(L, np.random.randn(T, dim))
 
noisy_trajectory = trajectory + epsilon

Why do we need to do the cholesky decomposition to get $L$? Why can’t we just use $K$?

Because the covariance