The following code block demonstrates how each of the faces can be approximately represented as a linear combination of only those 64 dominant eigenfaces. The inverse_transform() function from scikit-learn is used to go back to the original space but only with these 64 dominant eigenfaces, discarding all other eigenfaces:

# face reconstruction
faces_inv_proj = pipeline.named_steps['pca'].inverse_transform(faces_proj) 
#reshaping as 400 images of 64x64 dimension 
fig = plt.figure(figsize=(5,5)) 
fig.subplots_adjust(left=0, right=1, bottom=0, top=1, hspace=0.05, wspace=0.05) 
# plot the faces, each image is 64 by 64 dimension but 8x8 pixels 
j = 1
np.random.seed(0)
for i in np.random.choice(range(faces.shape[0]), 25):

    ax = fig.add_subplot(5, 5, j, xticks=[], yticks=[]) 
    ax.imshow(mean_face + sd_face*np.reshape(faces_inv_proj,(400,64,64)) [i,:], cmap=plt.cm.bone, interpolation='nearest')
    j += 1

The following screenshot shows the output of the preceding code block—25 randomly chosen face images reconstructed from the 64 eigenfaces. As can be seen, they look pretty much like the original faces (without many visible errors):

The following code block helps us to take a closer look and compares an original face with its reconstructed version:

orig_face = np.reshape(faces[0,:], (64,64))
reconst_face = np.reshape(faces_proj[0,:]@pipeline.named_steps['pca'].components_, (64,64))
reconst_face = mean_face + sd_face*reconst_face
plt.figure(figsize=(10,5))
plt.subplot(121), plt.imshow(orig_face, cmap=plt.cm.bone, interpolation='nearest'), plt.axis('off'), plt.title('original', size=20)
plt.subplot(122), plt.imshow(reconst_face, cmap=plt.cm.bone, interpolation='nearest'), plt.axis('off'), plt.title('reconstructed', size=20)
plt.show()

The following screenshot shows the output. As can be seen, the reconstructed face approximates the original face with some distortion: