Maybe it is just because I'm only experimenting with 2 dimensional normal distributions, but multi-dimensional normal distributions for me seem like just multiple one dimensional normal distributions.
However, in this Wikipedia link, the illustration just at the top right corner is a two dimensional normal distribution as well, but it defines covariance ($\sum$), instead of variance ($\sigma^2)$, and I don't understand why it's needed.
Built with this python code:
import matplotlib.pyplot as plt import numpy as np def gaussian(x, mean, variance): return 1 / (variance / np.sqrt(2 * np.pi)) * np.exp(-0.5 * (x - mean)**2 / variance**2) means = [4, 4] variances = [1, 0.8] placeholder = np.arange(0, 8, 0.01) epsilon = np.random.normal(size=(300, 2)) x = np.array(means) + np.exp(0.5 * np.array(variances)) * epsilon fig = plt.figure() fig = fig.gca(projection='3d') fig.plot(placeholder, gaussian(placeholder, means, variances), zs=8, zdir='y') fig.plot(placeholder, gaussian(placeholder, means, variances), zs=0, zdir='x') fig.scatter(x[:,0], x[:,1], zs = 0, zdir='z') fig.legend() fig.set_xlim(0, 8) fig.set_ylim(0, 8) fig.set_zlim(0, 4) fig.set_xlabel('X') fig.set_ylabel('Y') fig.set_zlabel('Z') plt.savefig('diagram.png')
I really need to understand this for my long journey to understand VAEs.