# Autoencoders and latent space fragmentation – V – reconstruction of human face images from simple statistical z-point-distributions?

In this post series we have so far studied the distribution of latent vectors and z-points for CelebA images in the latent space of an Autoencoder [AE]. The CelebA images show human faces. We want to reconstruct images with new faces from artificially created, statistical latent vectors. Our latent space had a number dimensions N=256. For basics of Autoencoders, terms like latent space, z-points, latent vectors etc. see the first blog of this series:

Autoencoders and latent space fragmentation – I – Encoder, Decoder, latent space

During the experiments and analysis discussed in the other posts

we have learned that the multi-dimensional latent space volume which the Encoder of a convolution AE fills densely with z-points for CelebA images has a special shape and location. The bulk of z-points is confined to a multi-dimensional ellipsoid which is relatively small. Its center has a position, which does not coincide with the latent space’s origin. The bulk center is located within or very close to a hyper-sub-volume spanned by only a few coordinate axes.

We also saw also that we have difficulties to hit this region by artificially created z-points via latent vectors. Especially, if the approach for statistical vector generation is a simple one. In the beginning we had naively assumed that we can treat the vector components as independent variables. We assigned each vector component values, which we took from a value-interval [-b, b] with a constant probability density. But we saw that such a simple and seemingly well founded method for statistical vector creation has a variety of disadvantages with respect to the bulk distribution of latent vectors for CelebA images. To name a few problems:

• The components of the latent vectors for CelebA images are correlated and not independent. See the third post for details.
• If we choose b too large (&gt. 2.0) then the length or radius values of the created vectors will not fit typical radii of the CelebA vectors. See the 2nd post for details.
• The generated vectors only correspond to z-points which fill parts of a thin multi-dimensional spherical shell within the cube filled by points with coordinate values -b < x_j < +b. This is due to mathematical properties of the distribution in multi-dimensional spaces. See the second post for more details.
• Optimal parameter values are 1.0 < b < 2.0, just to guarantee at least the right vector length. However, due to the fact that the origin of the latent space is located in a border region of the CelebA bulk z-point distribution, most points will end up outside the bulk volume.

In this post I will show you that our statistical vectors and the related z-points do indeed not lead to a successful reconstruction of images with clearly visible human faces. Afterward I will briefly discuss whether we still can have some hope to create new human face images by a standard AE’s Decoder and statistical points in its latent space.

# Relevant values of parameter b for the interval from which we choose statistical values for the vector components

What does the number distribution for the length of latent CelebA vectors look like? For case I explained in the 2nd post of this series we get:

For case II:

Now let me remind you of a comparison with the lengths of statistical vectors for different values of b:

See the 2nd post of this series for more details. Obviously, for our simple generation method used to create statistical vectors a parameter value b = 1.5 is optimal. b=1.0 an b=2.0 mark the edges of a reasonable range of b-values. Larger or smaller values will drive the vector lengths out of the required range.

# Image reconstructions for statistical vectors with independent component values x_j in the range -1.5 < x_j < 1.5

I created multiples of 512 images by feeding 512 statistical vectors into the Decoder of our convolutional Autoencoder which was trained on CelebA images (see the 1st and the 2nd post for details). The vector components x_j fulfilled the condition :

-1.5 ≤   x_j   ≤ 1.5,    for all j in [0, 256]

The result came out to be frustrating every time. Only singular image showed like outlines of a human face. The following plot is one example out of series of tenths of failures.

And this was an optimal b-value! 🙁 But due to the analysis of the 4th post in this series we had to expect this.

# Image reconstructions for statistical vectors with independent component values x_j in the range -1.0 < x_j < 1.0

The same for

-1.0 ≤   x_j   ≤ 1.0,    for all j in [0, 256]

# Image reconstructions for statistical vectors with independent component values x_j in the range -2.0 < x_j < 2.0

The same for

-2.0 ≤   x_j   ≤ 2.0,    for all j in [0, 256]

# Image reconstructions for statistical vectors with independent component values in the range -5.0 < x_j < 5.0

If you have read the previous posts in this series then you may think: Some of the component values of latent vectors for CelebA images had bigger values anyway. So let us try:

-1.0 ≤   x_j   ≤ 1.0,    for all j in [0, 256]

And we get:

Although some component values may fall into the value regions of latent CelebA vectors most of them do not. The lengths (or radii) of the statistical vectors for b=5.0 do not at all fit the average radii of latent vectors for CelebA images.

# What should we do?

I have described the failure to create reasonable images from statistical vectors in some regions around the origin of the latent space also in other posts on standard Autoencoders and in posts on Variational Autoencoders [VAEs]. For VAEs one enforces that regions around the origin are filled systematically by special Encoder layers.

But is all lost for a standard Encoder? No, not at all. It is time that we begin to use our knowledge about the latent z-point distribution which our AE creates for CelebA images. As I have shown in the 2nd post it is simple to get the number distribution for the component values. Because these distributions were similar to Gaussians we have rather well defined value regions per component which we can use for vector creation. I.e. we can perform a kind of first order approximation to the main correlations of the components and thus put our artificially created values inside the densely populated bulk of the z-point region for CelebA images. If that region in the latent space has a meaning at all then we should find some interesting reconstruction results for z-points within it.

We can do this even with our simple generation method based on constant probability densities, if we use individual value regions -b_j ≤ x_j ≤ b_j for each component. (Instead of a common value interval for all components). But even better would be Gaussian approximations to the real number distributions for the component values. In any case we have to restrict the values for each component much stronger than before.

# Conclusion

What we already had expected from previous analysis became true. A simple method for the creation of statistical latent vectors does not give us z-points which are good for the creation of reasonable images by a trained AE’s Decoder. The simplifications

1. that we can treat the component values of latent vectors as independent variables
2. and that we can assign the components x_j values from a common interval -b ≤ x_j &le b

cause that we miss the bulk region in the AE’s latent space, which gets filled by its trained Encoder. We have demonstrated this for the case of an AE which had been trained on images of human faces.

In the next post

Autoencoders and latent space fragmentation – VI – image creation from z-points along paths in selected coordinate planes of the latent space

I will show that even a simple vector creation can give us latent vectors within the region filled for CelebA images. And we will see that such points indeed lead to the reconstruction of images with clearly visible human face images by the Decoder of a standard AE trained on the CelebA dataset.

# Autoencoders and latent space fragmentation – III – correlations of latent vector components

The topics of this post series are

• convolutional Autoencoders,
• images of human faces, provided by the CelebA dataset
• and related data point and vector distributions in the AEs’ latent spaces.

In the first post

I have repeated some basics about the representation of images by vectors. An image corresponds e.g. to a vector in a feature space with orthogonal axes for all individual pixel values. An AE’s Encoder compresses and encodes the image information in form of a vector in the AE’s latent space. This space has many, but significantly fewer dimensions than the original feature space. The end-points of latent vectors are so called z-points in the latent space. We can plot their positions with respect to two coordinate axes in the plane spanned by these axes. The positions reflect the respective vector component values and are the result of an orthogonal projection of the z-points onto this plane. In the second post

Autoencoders and latent space fragmentation – II – number distributions of latent vector components

I have discussed that the length and orientation of a latent vector correspond to a recipe for a constructive process of The AE’s (convolutional) Decoder: The vector component values tell the Decoder how to build a superposition of elementary patterns to reconstruct an image in the original feature space. The fundamental patterns detected by the convolutional AE layers in images of the same class of objects reflect typical pixel correlations. Therefore the resulting latent vectors should not vary arbitrarily in their orientation and length.

By an analysis of the component values of the latent vectors for many CelebA images we could explicitly show that such vectors indeed have end points within a small coherent, confined and ellipsoidal region in the latent space. The number distributions of the vectors’ component values are very similar to Gaussian functions. Most of them with a small standard deviation around a central mean value very close to zero. But we also found a few dominant components with a wider value spread and a central average value different from zero. The center of the latent space region for CelebA images thus lies at some distance from the origin of the latent space’s coordinate system. The center is located close to or within a region spanned by only a few coordinate axes. The Gaussians define a multidimensional ellipsoidal volume with major anisotropic extensions only along a few primary axes.

In addition we studied artificial statistical vector distributions which we created with the help of a constant probability distribution for the values of each of the vector components. We found that the resulting z-points of such vectors most often are not located inside the small ellipsoidal region marked by the latent vectors for the CelebA dataset. Due to the mathematical properties of this kind of artificial statistical vectors only rather small parameter values 1.0 ≤ b ≤ 2.0 for the interval [-b, b], from which we pick all the the component values, allow for vectors with at least the right length. However, whether the orientations of such artificial vectors fit the real CelebA vector distribution also depends on possible correlations of the components.

In this post I will show you that there indeed are significant correlations between the components of latent vectors for CelebA images. The correlations are most significant for those components which determine the location of the center of the z-point distribution and the orientation of the main axes of the z-point region for CelebA images. Therefore, a method for statistical vector creation which explicitly treats the vector components as statistically independent properties may fail to cover the interesting latent space region.

# Normalized correlation coefficient matrix

When we have N variables (X_1, x_2, … x_n) and M parallel observations for the variable values then we can determine possible correlations by calculating the so called covariance matrix with elements Cij. A normalized version of this matrix provides the so called “Pearson product-moment correlation coefficients” with values in the range [0.0, 1.0]. Values close to 1.0 indicate a significant correlation of the variables x_i and x_j. For more information see e.g. the following links to the documentation on Numpy’s versions for the calculation of the (normalized) covariance matrix from an array containing the observations in an ordered matrix form: “numpy.cov” and to “numpy.corrcoef“.

So what are the “variables” and “observations” in our case?

# Latent vectors and their components

In the last post we have calculated the latent vectors that a trained convolutional AE produces for a 170,000 images of the CelebA dataset. As we chose the number N of dimensions of the latent space to be N=256 each of the latent vectors had 256 components. We can interpret the 256 components as our “variables” and the latent vectors themselves as “observations”. An array containing M rows for individual vectors and N columns for the component values can thus be used as input for Numpy’s algorithm to calculate the normalized correlation coefficients.

When you try to perform the actual calculations you will soon detect that determining the covariance values based on a statistics for all of the 170,000 latent vectors which we created for CelebA images requires an enormous amount of RAM with growing M. So, we have to chose M << 170,000. In the calculations below I took M = 5000 statistically selected vectors out of my 170,000 training vectors.

# Some special latent vector components

Before I give you the Pearson coefficients I want to remind you of some special components of the CelebA latent vectors. I had called these components the dominant ones as they had either relatively large absolute mean values or a relatively large half-width. The indices of these components, the related mean values mu and half-widths hw are listed below for a AE with filter numbers in the Encoder’s and Decoder’s 4 convolutional layers given by (64, 64, 128, 128) and (128, 128, 64, 64), respectively:

15   mu : -0.25 :: hw:  1.5
16   mu :  0.5  :: hw:  1.125
56   mu :  0.0  :: hw:  1.625
58   mu :  0.25 :: hw:  2.125
66   mu :  0.25 :: hw:  1.5
68   mu :  0.0  :: hw:  2.0
110   mu :  0.5  :: hw:  1.875
118   mu :  2.25 :: hw:  2.25
151   mu :  1.5  :: hw:  4.125
177   mu : -1.0  :: hw:  2.25
178   mu :  0.5  :: hw:  1.875
180   mu : -0.25 :: hw:  1.5
188   mu :  0.25 :: hw:  1.75
195   mu : -1.5  :: hw:  2.0
202   mu : -0.5  :: hw:  2.25
204   mu : -0.5  :: hw:  1.25
210   mu :  0.0  :: hw:  1.75
230   mu :  0.25 :: hw:  1.5
242   mu : -0.25 :: hw:  2.375
253   mu : -0.5  :: hw:  1.0

The first row provides the component number.

# Pearson correlation coefficients for dominant components of latent CelebA vectors

For the latent space of our AE we had chosen the number N of its dimensions to be N=256. Therefore, the covariance matrix has 256×256 elements. I do not want to bore you with a big matrix having only a few elements with a size worth mentioning. Instead I give you a code snippet which should make it clear what I have done:

import numpy as np
#np.set_printoptions(threshold=sys.maxsize)

# The Pearson correlation coefficient matrix
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
print(z_points.shape)
print()
num_pts      = 5000

# Special points in slice
num_pts_spec = 100000
jc1_sp = 118; jc2_sp = 164
jc1_sp = 177; jc2_sp = 195

len_z = z_points.shape[0]

ay_sel_ptsx = z_points[np.random.choice(len_z, size=num_pts, replace=False), :]
print(ay_sel_ptsx.shape)

# special points
threshcc = 2.0
ay_sel_pts1 = ay_sel_ptsx[( abs(ay_sel_ptsx[:,:jc1_sp])         < threshcc).all(axis=1)]
print("shape of ay_sel_pts1 :  ", ay_sel_pts1.shape )
ay_sel_pts2 = ay_sel_pts1[( abs(ay_sel_pts1[:,jc1_sp+1:jc2_sp]) < threshcc).all(axis=1)]
print("shape of ay_sel_pts2 :  ", ay_sel_pts2.shape )
ay_sel_pts3 = ay_sel_pts2[( abs(ay_sel_pts2[:,jc2_sp+1:])       < threshcc).all(axis=1)]
print("shape of ay_sel_pts3 :  ", ay_sel_pts3.shape )
ay_sel_pts_sp  = ay_sel_pts3

ay_sel_pts = ay_sel_ptsx.transpose()
print("shape of ay_sel_pts :  ", ay_sel_pts.shape)

ay_sel_pts_spec = ay_sel_pts_sp.transpose()
print("shape of ay_sel_pts_spec :  ",ay_sel_pts_spec.shape)
print()

# Correlation corefficients for the selected points
corr_coeff = np.corrcoef(ay_sel_pts)
nd = corr_coeff.shape[0]

print(corr_coeff.shape)
print()

for k in range(1,7):
thresh = k/10.
print( "num coeff >", str(thresh), ":", int( ( (np.absolute(corr_coeff) > thresh).sum() - nd) / 2) )

The result was:

(170000, 256)

(5000, 256)
shape of ay_sel_pts1 :   (101, 256)
shape of ay_sel_pts2 :   (80, 256)
shape of ay_sel_pts3 :   (60, 256)
shape of ay_sel_pts :   (256, 5000)
shape of ay_sel_pts_spec :   (256, 60)

(256, 256)

num coeff > 0.1 : 1456
num coeff > 0.2 : 158
num coeff > 0.3 : 44
num coeff > 0.4 : 25
num coeff > 0.5 : 16
num coeff > 0.6 : 8

The lines at the end give you the number of pairs of component indices whose correlation coefficients are bigger than a threshold value. All numbers vary a bit with the selection of the random vectors, but in narrow ranges around the values above. The intermediate part reduces the amount of CelebA vectors to a slice where all components have small values < 2.0 with the exception of 2 special components. This reflects z-points close to the plane panned by the axes for the two selected components.

Now let us extract the component indices which have a significant correlation coefficient > 0.5:

li_ij = []
li_ij_inverse = {}
# threshc  = 0.2
threshc  = 0.5

ncc = 0.0
for i in range(0, nd):
for j in range(0, nd):
val = corr_coeff[i,j]
if( j!=i and abs(val) > threshc ):
# Check if we have the index pair already
if (i,j) in li_ij_inverse.keys():
continue
# save the inverse combination
li_ij_inverse[(j,i)] = 1
li_ij.append((i,j))
print("i =",i,":: j =", j, ":: corr=", val)
ncc += 1

print()
print(ncc)
print()
print(li_ij)

We get 16 pairs:

i = 31  :: j = 188 :: corr= -0.5169590614268832
i = 68  :: j = 151 :: corr=  0.6354094560888554
i = 68  :: j = 177 :: corr= -0.5578352818543628
i = 68  :: j = 202 :: corr= -0.5487381785057351
i = 110 :: j = 188 :: corr=  0.5797971250208538
i = 118 :: j = 195 :: corr= -0.647196329744637
i = 151 :: j = 177 :: corr= -0.8085621658509928
i = 151 :: j = 202 :: corr= -0.7664405924287517
i = 151 :: j = 242 :: corr=  0.8231503928254471
i = 177 :: j = 202 :: corr=  0.7516815584868468
i = 177 :: j = 242 :: corr= -0.8460097558498094
i = 188 :: j = 210 :: corr=  0.5136571387916908
i = 188 :: j = 230 :: corr= -0.5621165900366926
i = 195 :: j = 242 :: corr=  0.5757354150766792
i = 202 :: j = 242 :: corr= -0.6955230633323528
i = 210 :: j = 230 :: corr= -0.5054635808381789

16

[(31, 188), (68, 151), (68, 177), (68, 202), (110, 188), (118, 195), (151, 177), (151, 202), (151, 242), (177, 202), (177, 242), (188, 210), (188, 230), (195, 242), (202, 242), (210, 230)]

You note, of course, that most of these are components which we already identified as the dominant ones for the orientation and lengths of our latent vectors. Below you see a plot of the number distributions for the values the most important components take:

# Visualization of the correlations

It is instructive to look at plots which directly visualize the correlations. Again a code snippet:

import numpy as np
num_per_row = 4
num_rows    = 4
num_examples = num_per_row * num_rows

li_centerx = []
li_centery = []
li_centerx.append(0.0)
li_centery.append(0.0)

#num of plots
n_plots = len(li_ij)
print("n_plots = ", n_plots)

plt.rcParams['figure.dpi'] = 96
fig = plt.figure(figsize=(16, 16))

#special CelebA point
n_spec_pt = 90415

# statisitcal vectors for b=4.0
delta = 4.0
num_stat = 10
ay_delta_stat = np.random.uniform(-delta, delta, size = (num_stat,z_dim))

print("shape of ay_sel_pts : ", ay_sel_pts.shape)

n_pair = 0
for j in range(num_rows):
if n_pair == n_plots:
break
offset = num_per_row * j
# move through a row
for i in range(num_per_row):
if n_pair == n_plots:
break
j_c1 = li_ij[n_pair][0]
j_c2 = li_ij[n_pair][1]
li_c1 = []
li_c2 = []
for npl in range(0, num_pts):
#li_c1.append( z_points[npl][j_c1] )
#li_c2.append( z_points[npl][j_c2] )
li_c1.append( ay_sel_pts[j_c1][npl] )
li_c2.append( ay_sel_pts[j_c2][npl] )

# special CelebA point
li_spec_pt_c1=[]
li_spec_pt_c2=[]
li_spec_pt_c1.append( z_points[n_spec_pt][j_c1] )
li_spec_pt_c2.append( z_points[n_spec_pt][j_c2] )

# statistical vectors
li_stat_pt_c1=[]
li_stat_pt_c2=[]
for n_stat in range(0, num_stat):
li_stat_pt_c1.append( ay_delta_stat[n_stat][j_c1] )
li_stat_pt_c2.append( ay_delta_stat[n_stat][j_c2] )

# plot
sp_names = [str(j_c1)+' - '+str(j_c2)]
axc = fig.add_subplot(num_rows, num_per_row, offset + i +1)
#axc.axis('off')
axc.scatter(li_c1, li_c2, s=0.8 )
axc.scatter(li_stat_pt_c1, li_stat_pt_c2, s=20, color="red", alpha=0.9 )
axc.scatter(li_spec_pt_c1, li_spec_pt_c2, s=80, color="black" )
axc.scatter(li_spec_pt_c1, li_spec_pt_c2, s=50, color="orange" )
axc.scatter(li_centerx, li_centery, s=100, color="black" )
axc.scatter(li_centerx, li_centery, s=60, color="yellow" )
n_pair += 1

The result is:

The (5000) blue dots show the component values of the randomly selected latent vectors for CelebA images. The yellow dot marks the origin of the latent space’s coordinate system. The red dots correspond to artificially created random vectors for b=4.0. The orange dot marks the values for one selected CelebA image. We also find indications of an ellipsoidal form of the z-point region for the CelebA dataset. But keep in mind that we only a re looking at projections onto planes. Also watch the different scales along the two axes!

# Interpretation

The plots clearly show some average correlation for the depicted latent vector components (and their related z-points). We also see that many of the artificially created vector components seem to lie within the blue cloud. This appears a bit strange as we had found in the last post that the radii of such vectors do not fit the CelebA vector distribution. But you have to remember that we only look at projections of the real z-points down to some selected 2D-planes within of the multi-dimensional space. The location in particular projections does not tell you anything about the radius. In a later sections I also show you plots where the red dots quite often fall outside the blue regions of other components.

I want to draw your attention to the fact that the origin seems to be located close to the border of the region marked by some components. At least in the present projection of the z-points to the 2D-planes. If we only had the plots above then the origin could also have a position outside the bulk of CelebA z-points. The plots confirm however what we said in the last post: The CelebA vector distributions has its center off the origin.

We also see an indication that the density of the z-points drops sharply towards most of the border regions. In the projections this becomes not so clear due to the amount of points. See the plot below for only 500 randomly selected CelebA vectors and the plots in other sections below.

# Border position of the origin with respect to the latent vector distribution for CelebA

Below you find a plot for 1000 randomly selected CelebA vectors, some special components and b=4.0. The components which I selected in this case are NOT the ones with the strongest correlations.

These plots again indicate that the border position of the latent space’s origin is located in a border region of the CelebA z-points. But as mentioned above: We have to be careful regarding projection effects. But we also have the plot of all number distributions for the component values; see the last post for this. And there we saw that all the curves cover a range of values which includes the value 0.0. Together we the plots above this is actually conclusive: The origin is located in a border region of the latent z-point volume resulting from CelebA images after the training of our Autoencoder.

This fact also makes artificial vector distributions with a narrow spread around the origin determined by a b ≤ 2.0 a bit special. The reason is that in certain directions the component value may force the generated artificial z-point outside the border of the CelebA distribution. The range between 1.0 < b < 2.0 had been found to be optimal for our special statistical distribution. The next plot shows red dots for b=1.5.

This does not look too bad for the selected components. So we may still hope that our statistical vectors may lead to reconstructed images by the Decoder which show human faces. But note: The plots are only projections and already one larger component-value can be enough to put the z-point into a very thinly populated region outside the main volume fo CelebA z-points.

# Conclusion

The values for some of the components of the latent vectors which a trained convolutional AE’s Encoder creates for CelebA images are correlated. This is reflected in plots that show an orthogonal projection of the multi-dimensional z-point distribution onto planes spanned by two coordinate axes. Some other components also revealed that the origin of the latent space has a position close to a border region of the distribution. A lot of artificially created z-points, which we based on a special statistical vector distribution with constant probabilities for each of the independent component values, may therefore be located outside the main z-point distribution for CelebA. This might even be true for an optimal parameter b=1.5, which we found in our analysis in the last post.

We will have a closer look at the border topic in the next post:

Autoencoders and latent space fragmentation – IV – CelebA and statistical vector distributions in the surroundings of the latent space origin

# Autoencoders and latent space fragmentation – II – number distributions of latent vector components

This post series studies the (in-) ability of a trained Autoencoder [AE] to create reasonable human face images from statistical vectors placed in its latent space. In my last post

Autoencoders and latent space fragmentation – I – Encoder, Decoder, latent space

I have described what the purposes of the two sub-networks of an AE, i.e. the Encoder and the Decoder, are. We saw that the so called latent space plays an important role for the interplay of these sub-networks: Vectors in the latent space – z-pointsencode properties of objects presented to the Autoencoder, more precisely to its Encoder. The bunch of objects used during training thus gives us a distribution of vectors and respective z-points within certain regions of the latent space. The Decoder reconstructs objects from latent vectors.

One of my eventual objectives in this series is the creation of new objects of the same class presented to the AE during its training. I focus on the special case of images displaying human faces. Therefore, I have trained a convolutional AE with the so called CelebA dataset. After training one may hope that an AE will be able to produce images with new faces, which are not present in CelebA, when we feed the Decoder with suitable z-points. The question is what “suitable z-points” are and where they are located in the latent space.

Of course, I want to use the Decoder’s reconstruction abilities to achieve my goal. To get new faces, a statistical element is a must. The basic idea is to use statistically created z-points as input for the Decoder.

# Objective of this post

In the first post of this series I have already indicated that not all vectors in the AE’s latent space may lead to the production of reasonable images. It might well be that we must hit certain confined regions of the latent space. An interesting question, therefore, is the following:

Are all generators of statistical vectors suitable to hit the regions of a latent space which an Autoencoder will fill for certain training objects?

In this and further posts I want to show you that this is not the case. A bad choice of a statistical generating method in combination with the high number of latent space dimensions may lead to a complete failure.

To achieve this insight we must study both the real z-point distribution which an AE creates for CelebA images and the artificial vector or z-point distribution coming from a specific statistical generator. In this post I study a generator which assigns the vector components values which are taken from a real number interval with a constant probability. The number of dimensions N of the latent space shall be N=256.

We shall see that we indeed get confronted with a special side of the curse of high dimensionality and that our artificial z-point distribution does not match the real one for CelebA at all if we do not restrict parameters in a somewhat counter-intuitive way. As a side effect we will learn that our AE organizes the latent vector distributions for CelebA images via functions very similar to Gaussians. Furthermore, we shall see that we speak of just one coherent and confined z-point region with very small extensions. The center of this region sits close to a hypervolume spanned by only a few (out of 256) coordinate axes.

# Methods to create statistical z-points

To create statistical z-points in a latent space we have to employ some generating mechanism for respective statistical vectors. See my first post for the correspondence of z-points to latent vectors. There are multiple ways available to create latent vectors. I just name 3 popular ones:

1. Create a dense homogeneous distribution by filling some volume around the origin with a grid of points.
2. Use a constant probability density for values in a real number interval [-b, b], pick such values statistically and assign them to individual vector components.
3. Use one or multiple Gaussian distributions to define the vector component values.

Note that when applying the second and the third method the statistics works on the level of the vector components. These components are handled as independent variables, each obeying a certain probability distribution of assignable real values.

A small calculation shows that method 1 will not work in practice, as such a distribution requires an enormous number of points for high dimensions and a decent resolution. Method 2 looks simple and works well in 2D- and 3D-spaces. The third method requires assumptions about the mean values and standard deviations to take – per vector coordinate.

Therefore it is tempting to use method 2 to create one or more artificial statistical distribution of random vectors in the latent space. This is exactly what we will try in this post – in the hope that a significant part of the resulting points will hit regions which give us reasonable images.

If you are interested in mathematical properties of vector distributions created by method 2 in multi-dimensional spaces, you will find them in the following posts of this blog:

# Why must we hit specific regions of the latent space?

Experience tells us that we will not get reasonable images from arbitrarily and randomly placed z-points in the latent space.. (Concrete examples will be given in the next post.) What could a plausible reason be?

Convolutional networks [CNNs] extract patterns and save them in parameters of their layer filters (or neural maps). In another post series I have shown such elementary patterns, which the innermost convolutional layers react sensitively to, for the simple case of MNIST. Patterns correspond to correlations between constituting elements of the objects we present to the neural networks. Such patterns reflect certain features of the objects. The number of elementary patterns a CNN can handle depends on the number of available kernel filters – which is fixed by the network structure. For a trained convolutional Autoencoder it is therefore reasonable to assume that a latent space vector encodes a prescription for the superposition of certain elementary patterns by which the Decoder eventually creates an image. The information for the pattern mixture is encoded both by the length and angles of latent vectors. The latter with respect to the many coordinate axes; the multitude of angles describes the orientation of a latent vector in its multi-dimensional space.

It is clear that not all prescriptions for a mixture of elementary patterns will reflect the real pixel correlations of a human face in front of some background. We must therefore assume that the latent space regions filled by a trained AE-Encoder for the training objects are the ones which give us reasonable Decoder results. In principle we could find multiple such regions in different parts of the latent space. They could have particular locations and could be confined to a relatively small volume. Therefore, we should really check that at least a part of our statistical vectors points to those regions.

# Number distributions for vector components

A priori we do not know anything about the shape of z-point distribution which an Autoencoder will create in its latent space for CelebA data. Therefore, we need to get an overview over some properties of such a z-point distribution. As multidimensional spaces with a high number of dimensions like N ≥ 256 can not be presented in 3D, we need some other kind of visualization. What we shall use is a kind of spectral display for the coordinate values of our vectors:

We are going to analyze the number distribution for the values of each of the vector components.

I.e., we count the numbers of latent vectors which have values for a certain component in a series of sampling intervals for real values. We do this for all components and for a reasonable total range of values. Note that a distribution for a specific component is a one-dimensional function over ℜ.

Below we will first derive the component related distributions from real Autoencoder vectors for CelebA images. This will tell us already a lot about the orientation, the off-center location and the extensions of the real z-point distribution for CelebA.

Afterward, we will compare the CelebA specific distributions to the number distributions for artificial statistical vector distributions created by method 2.

# Number distributions for vector lengths

Another nice and simple method to analyze the compatibility of vector distributions is to compare the number distributions for the vector lengths. For an orthogonal coordinate system of the latent space we can compute the length of a multi-dimensional vector by an Euclidean L2-norm. We will compare the number distribution for the lengths of latent CelebA vectors with the distribution for the lengths of vectors created by method 2. We will call the length of a vector also its “radius”.

# Network setup

The basic layer structure of my AE was described already in the last post. It is relatively simple. I employ only a few Encoder and Decoder layers. I have ensured the Encoder and Decoder are actually able to solve the basic task of encoding and decoding data of real CelebA images.

We look at the results of two AE networks which differ in the number of convolutional kernel filters used:

• Test case I: We use 4 Conv2D layers in the Encoder with 32, 64, 128, 256 filters and 4 TransposeConv2D layers in the Decoder with 256, 128, 64, 32 filters, respectively.
• Test case II: We use 4 Conv2D layers in the Encoder with 64, 64, 128, 128 filters and 4 TransposeConv2D layers in the Decoder with 128, 128, 64, 64 filters, respectively.

This is reflected in the following code snippet. There you also get information on the kernel sizes, strides and padding-methods. The number of dimensions of the latent space is N = z_dim = 256. The activation function is a chosen to be Leaky Relu.

# Test case I
AE1 = Autoencoder(
input_dim                  = INPUT_DIM
, encoder_conv_filters     = [32,64,128,256]       # We take a bit bigger than D. Foster
, encoder_conv_kernel_size = [3,3,3,3]
, encoder_conv_strides     = [2,2,2,2]

, decoder_conv_t_filters     = [128,64,32,n_ch]    # !!! n_ch = 1 or 3
, decoder_conv_t_kernel_size = [3,3,3,3]
, decoder_conv_t_strides     = [2,2,2,2]
, z_dim = 256
, act   = 0                  # activation 0:Leaky ReLU (standard), 1: ReLU, 2: SELU
)
# test case II
AE2 = Autoencoder(
input_dim                  = INPUT_DIM
, encoder_conv_filters     = [64,64,128,128]       # We take a bit bigger than D. Foster
, encoder_conv_kernel_size = [5,5,3,3]
, encoder_conv_strides     = [2,2,2,2]

, decoder_conv_t_filters     = [128,64,64,n_ch]    # !!! n_ch = 1 or 3
, decoder_conv_t_kernel_size = [3,3,5,5]
, decoder_conv_t_strides     = [2,2,2,2]
, z_dim = 256
, act   = 0                  # activation 0:Leaky ReLU (standard), 1: ReLU, 2: SELU
)

# A method to analyze the vector distribution in a high-dimensional vector space

The components of our latent vectors determine their angle and length. We base our analysis of corresponding z-points on the number distribution per component-value. To do this we select a suitable real value interval covering all the values for vector components which the AE actually uses. We divide this interval into a series of sufficient sub-intervals for data sampling. In our case around 100 sub-intervals.

After training of our AE we once again feed all training objects (in our case > 170,000 CelebA images) into the Encoder and keep the vectors in some Numpy arrays. Then we look at a specific component and a sampling interval and count the number of vectors for which the component value resides inside the sampling interval. Repeating this for all components and intervals we get a number distribution which can be plotted. If we are lucky the resulting shapes of the number distributions will give us information about the corresponding shape of the multi-dimensional regions which the AE fills for CelebA images.

# CelebA images: Number distribution for component values of latent vectors

The following plot shows the number distributions for all of the 256 components of vectors for CelebA images in our trained AE’s latent space:

Case I: Number distribution after 24 epochs

Case I: Selected components

Case II: Number distribution after 30 epochs

Case II: Selected components

We see that the individual number distributions are very similar to Gaussian distributions. For test case II I also give you the values for the central average value μ (named mu in the list below) and the half-width (named hw below) of the most interesting components. The half-width is the difference between those coordinate values where the distribution function achieves a value of half of the maximum number value at μ:

15 mu : -0.25 :: hw:  1.5
16 mu :  0.5  :: hw:  1.125
56 mu :  0.0  :: hw:  1.625
58 mu :  0.25 :: hw:  2.125
66 mu :  0.25 :: hw:  1.5
68 mu :  0.0  :: hw:  2.0
110 mu :  0.5  :: hw:  1.875
118 mu :  2.25 :: hw:  2.25
151 mu :  1.5  :: hw:  4.125
177 mu : -1.0  :: hw:  2.25
178 mu :  0.5  :: hw:  1.875
180 mu : -0.25 :: hw:  1.5
188 mu :  0.25 :: hw:  1.75
195 mu : -1.5  :: hw:  2.0
202 mu : -0.5  :: hw:  2.25
204 mu : -0.5  :: hw:  1.25
210 mu :  0.0  :: hw:  1.75
230 mu :  0.25 :: hw:  1.5
242 mu : -0.25 :: hw:  2.375
253 mu : -0.5  :: hw:  1.0

These components obviously have either a relatively large absolute μ-value or a relatively large half-width. I call these components the dominant ones.

# Interpretation of the number distributions per vector component

The first thing these plots prove is the fact that the results for different networks are different, too. Without proving it, I also say from experience that even two different training runs for one and the same AE network structure may result in somewhat different number distributions. But although there are some differences there are also striking similarities:

1. Most of the components show a Gaussian like number distribution about some mean value.
2. The mean coordinate value μ for most of the components (≥ 90%) is zero or close to it.
3. The component values cover a region of [-12, 12] in our specific case.
4. There are only a few components ( ≤ 25) with mean values <μ> ≥ 0.25 or <μ> ≤ -0.25.
5. There are only a few components ( ≤ 10) with mean values <μ> ≥ +1.0 or <μ> ≤ -1.
6. There are only relatively few components (≤ 40) with a half-width ≥ 1.
7. There are only a few components (≤ 20) with a half-width ≥ 1.5
8. There are only very few components ≤ 5 with a half-width ≥ 3.

A bit of thinking and imagination tells us that the center of the distribution must be located somewhat off the origin, but very close to or within a hypervolume spanned by only a few dominant coordinate axes. The multi-dimensional region filled by the z-points has significant anisotropic extensions or elongations around its center only in a few directions of the multidimensional space.

All in all we speak about a very specific, limited multi-dimensional region, located at some distance from the origin (but not too far) with a center point close to or within a sub-volume of very low dimensions spanned by some coordinate axes. The overall direction of the center with respect to the origin is well defined by a few coordinates. The diameters of the regions are small in most directions. The z-points concentrate strongly towards the center of this region. Significant diameters around the center are only given in some particular directions. The respective axes involved are less than 10% of the total number.

This also means that a primary component analysis of the z-point distribution should only give us a number of dominant main components in the same number region (0.1 * N). See forthcoming posts for such an analysis.

Analogon in 3D: In an analogous case within a 3-dimensional space we would speak of a kind of ellipsoid with a significant diameter beyond 1 only in a specific direction and a center located close to a line in a plane spanned by 2 of the 3 coordinate axes.

# Comparison to the number distribution for statistically created vectors with a constant probability for component values in a specific interval

Now we look at the number distributions for all components of 200,000 vectors created by our method 2. We choose the same region for values [-12, 12] as displayed for our test cases I and II above. The following plot confirms what you certainly have already guessed:

This plot just reflects the design of our probability distribution for each of the components. But the plot also indicates clearly that most of our statistically created points will not hit the latent space region which is filled by the Autoencoder for CelebA data.

# The statistics obviously plays against us

It is very instructive to write a small program which scans all of the artificially created vectors and checks whether their end points lie within the region defined by the real CelebA points. I leave this to the reader. To get a good guess I personally defined a region of three times the half-width left and right of each component value center of the real distribution. The number of points fulfilling all criteria for our CelebA region came out to be exactly zero. Even with 12 times the half-width you get only very few vectors pointing into the CelebA region (around 20 vectors). Why is this the case?

# The curse of a high dimensionality and a constant probability distribution

The first point is that for some components of the CelebA vectors the half-width really is rather small. So you do not cover the whole value interval [-12, 12] for the component values, but only around 75% for 12 times the half-width. Let us assume that this is the case for 7 to 8 components. Let us further assume that another 50 components the width is 90%. Then the probability to get a point is (0.75)8 * 0.965 ≈ 0.1 * 0.0018 ≈ 0.00018. This gives us only 30 out of 170,000 vectors potentially fulfilling our conditions. We are in that range.

But we know already that 90% of all components should hit an interval [-3,3]. The probability for this in the case of our method 2 is 0.25 per component. The probability for a hit thus is 0.25220 which is zero for all practical purposes. This is it what really kills our efforts to place a point in the most interesting regions of the latent space with the help of method 2 and seemingly fitting values of b.

Now, you may think: A decisive parameter for method 2 ist the interval [-b, b] from which I pick my statistical component values. What if we diminish b? E.g. to b=3 or b=2? This is a good idea as we shall see in the next paragraph. Yet, you still get only a few vectors (< 10) fulfilling all criteria for two times the half-width.

Let us reduce our b-interval to [-3, 3] for vector creation. Then for a typical vector more than 50 components miss the target region. Test runs show that even for [-2, 2] only around 10 out of 170.000 vectors would hit (outer parts) of the target region for CelebA images. In this case the few components which have off-center mu values seemingly work against us.

Things change dramatically for b=1.5 and 2 times the half-width. Now the components of all the artificial vectors fulfill our criteria. If you, however, change the intervals to hit to 1.5 times the half-width you are back again to only a very few vectors (< 10). For b=1 and 1.5 times the half-width we get again a number of 170,000 vectors fulfilling our criteria.

Why does this happen? And does it mean that when we restrict our component values to [-1.5, 1.5] we would cover our real CelebA distribution?

# Number distributions for the vector lengths

Below you find a plot showing the number distribution with respect to typical vector lengths – both for CelebA (in red) and vectors artificially created with method 2 (other colors).

The parameter “b” of our artificial distributions defines the interval [-b, b] from which we pick component values. The probability density is a constant for this interval.

The first point you may dwell upon is the fact that the radius values get so big – much bigger than b. This is due to the high number of dimensions; see the posts named for a mathematical cover of method 2.

The second counter-intuitive point may be the following: One expects that b=12 should really be a reasonable parameter value for method 2 to cover the range of component values for CelebA. But already for b=3 or b=4 we get vectors outside the vector length interval which CelebA latent vectors fill. The reason for this are properties of the artificial radius distributions which can be derived mathematically. A mathematical calculation of the expectation value for the mean length (= radius R) of our artificially generated vectors gives us a value around

<R>    ≈    b * sqrt(1/3 * N) * sqrt( 1 / (1 + 1/(4*N) )

with a relatively very narrow spread. See the derivation in the posts quoted above. sqrt in the formula above stands for the square root. The standard deviation Δstd(R) has a size of approximately

Δstd(R)    ≈    b * sqrt(1/15 * N) * sqrt( 1 + 1/(4*N) )

The ratio of the half-width to radius thus declines with the square root of the number of dimensions N. As we see, only the artificial distributions for b=1, b=1.5, b=2 cover parts of the radius distribution for latent CelebA vectors.

So, if we want to use method 2 then we should work with b-values 1.0 ≤ b ≤ 2.0 to get a probability > 0 for creating reasonable images of human faces.

# Will a proper value of b guarantee us reasonable face images?

We have seen that b must be reduced to a range 1.0 ≤ b ≤ 2.0 to get reasonable radius values of our statistical vectors. Unfortunately, this does not guarantee us proper images either. The reason is that there might be correlations between the dominant component values which our simple number distributions do not reveal. We will take care of this in the next post. For now I just show you a plot of the correlation between two specific dominant components for 1000 randomly selected CelebA latent vectors:

# Conclusion

In this post we have partially analyzed the distribution of vectors and related z-points which an Autoencoder creates for CelebA images in its latent space. We have found that the number distributions per vector component look like Gaussian distributions. While most of the components have a small spread around the value zero, there are a few dominant components which determine the (off-center) location and orientation of a coherent, confined and ellipsoidally shaped region for CelebA z-points. The center of this region is close to a hypervolume defined by a few axes.

It was a bit counter-intuitive to see that a simple method to create statistical z-points via a constant probability distribution for individual component values would obviously miss the relevant latent region for CelebA images totally. We saw that we would need very special parameter values to limit the component values to get artificial latent vectors with the required length. These findings alone make it very improbable that arbitrary z-points created without some restrictions for their component values would lead to reasonable face image creation by the Decoder.

Something that we have not yet covered is the question of correlations between vector components. This is the topic of the next post: