I'm trying to understand figure 12.1 in Goodfellow available here. I'm not able to reproduce figure 12.1, and I'm wondering what is it I'm missing. The denominator of equation 12.3 is a constant, and thus equation 12.3 reduces to a subtraction and a scaling. I'm finding it hard to believe that it will map the points to a sphere/circle as shown in figure 12.1. I'd expect something non-linear in order to do that. What am I missing?
My code is:
import numpy as np
import matplotlib.pyplot as plt
nn = 32
sigma = 1.0
mu = 0
def global_contrast_normalization(X,eps,lamb,ss):
mean_ = np.mean(X)
Xnew = X - mean_
ni,nj,nk = X.shape
denom = (1/(ni*nj*nk))*np.sum((X-mean_)**2)
denom = lamb + denom
denom = np.sqrt(denom)
denom = np.maximum(eps,denom)
Xnew = (ss*Xnew)/denom
return Xnew
img = np.random.normal(mu,sigma,size=(nn,nn,2))
imgnew = global_contrast_normalization(img,eps=1e-8,lamb=0,ss=1.0)
fig,(ax1,ax2) = plt.subplots(nrows=1,ncols=2)
ax1.set_aspect('equal')
ax2.set_aspect('equal')
pts1 = img.reshape((nn*nn,2))
ax1.scatter(pts1[:,0],pts1[:,1])
ax1.set_title('Data')
pts2 = imgnew.reshape((nn*nn,2))
ax2.scatter(pts2[:,0],pts2[:,1])
ax2.set_title('After GCN')
