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Autoencoders and latent space fragmentation – IX – PCA transformation of the z-point distribution for CelebA

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This series of posts is about standard convolutional Autoencoders [AEs]. We wish to use the creative ability of an AE’s Decoder for the creation of human face images. We trained a relatively simple example AE on the CelebA data set. For the generation of new human face images we wanted to feed the Decoder with randomly created z-vectors in the AE’s latent space. As arbitrary latent vectors did not deliver reasonable images, we had a look at properties of the z-point distribution created in the latent space after training. And we found the z-vectors with end-points within this particular region indeed gave us some first reasonable images.

But during the first posts we also had to learn that it requires substantial effort to produce “randomly” created z-vectors which point to the coherent and confined latent space region populated by z-points for CelebA images. In addition we found that we obviously must fulfill complex correlation conditions between the latent vector components. In the last posts we have, therefore, investigated the structure of the latent space region populated for CelebA images a bit more closely. See:

Autoencoders and latent space fragmentation – VIII – approximation of the latent vector distribution by a multivariate normal distribution and ellipses

Autoencoders and latent space fragmentation – VII – face images from statistical z-points within the latent space region of CelebA

This gave us a somewhat surprising insight:

For our special case of human face images the density distribution of the z-points in the multidimensional latent space roughly resembled a multivariate normal distribution. The variables of this distribution were, of course, the components of the latent z-vectors (reaching from the origin of the latent space coordinate system to the z-points). The number density distributions for the values of the z-vector components could very well be approximated by Gaussian functions. In addition we found strong linear correlations between the z-vector components. This was completely consistent with diagonally oriented elliptic contour lines of the number density distribution for pairs of z-vector components in the respective coordinate planes. The angles between the ellipses’ axes and the axes of the (orthogonal) coordinate system varied.

In this post we test my thesis of a multivariate normal distribution in a different and more complicated way. If the number density distribution of the z-points for CelebA really is close to a multivariate normal distribution then we would expect that the contour lines of the 2-dimensional number density function for coordinate planes still are ellipses after a PCA transformation. Note that the coordinate planes then are planes of a new coordinate system (with orthogonal axes) moved and rotated with respect to the original coordinate system. We call the transformed coordinate system the “PCA coordinate system“. (It is clear the vector components must consistently be transformed with respect to the new coordinate axes.)

This test is much harder because asymmetries, imperfections and deviations from a normal distribution will become clearly visible. Nevertheless we shall see that the z-point distribution decomposes roughly in uncorrelated Gaussian number density (or probability) functions for the (transformed) components of the z-vectors – as expected.

Note that number density distributions can be interpreted as probability distributions after a proper normalization. To understand the results and implications of this post some basic knowledge about multivariate normal probability density distributions in multidimensional spaces is required. Transformation properties of such distributions are important. In particular the effect of a PCA or SVD transformation, which diagonalizes the Pearson correlation matrix, on a multivariate normal distribution, should be familiar: In the moved and rotated coordinate system original linear correlations between vector components disappear. The coordinate axes of the transformed coordinate system get aligned to the eigenvectors (and main axes) of the probability density distribution. In addition elliptic contour lines prevail and the main axes of the ellipses get aligned with the axes of the PCA coordinate system.

For valuable information in previous posts see the link list a the bottom of this article.

Imperfections in the multivariate normal distribution and their consequences for PCA

In our case the latent space had 256 dimensions. The real number density functions for the values of the (256) latent vector components did not always show the full symmetry of their Gaussian fits. Deviations appeared near the center and at the flanks of the individual distributions. Still we got convincing overall ellipses in contour plots of the probability density for selected pairs of components. But there were also relatively many points which lay in regions outside the 2 to 3 sigma confidence ellipses for these 2-dimensional distributions – and the distribution of the points there seemed to violate the elliptic symmetry.

A perfect multivariate normal distribution (with only linear correlations between its variables) decomposes into (seemingly) uncorrelated normal distributions for the new variables, namely the latent vector components in the transformed coordinate system of the latent space. The coordinate transformation corresponds to a diagonalization of the Pearson correlation matrix via finding orthogonal eigenvectors. So we should not only get ellipses after the PCA transformation. The main axes of the resulting ellipses should also be aligned with the coordinate axes of the new translated and rotated PCA coordinate system. (This is a standard feature of “uncorrelated” Gaussians).

Note that the de-correlation is a pure coordinate effect which does not eliminate the dependencies of the original variables. The new coordinates and the respective variables do not have the same meaning as the old ones. We “only” change perspectives such that the mathematical description of the multivariate distribution gets simpler. However, the ratios of the axes of the new ellipses depend on the axis-ratios of the old ellipses in the original standard coordinate system of the latent space.

When we translate and rotate a coordinate system to diagonalize the Pearson matrix of an imperfect multivariate normal distribution the coordinate axes may afterward not fully align with the main axes of the distribution’s inner core – and then we might get clearly visible asymmetries in z-point projections to coordinate planes. Note that points outside the inner core have a relatively large impact on the PCA transformation and especially on the rotation of the new axes with respect to the old ones.

Reduction of outer z-point contributions

Outer z-points often correspond to CelebA images with strong variations in the background. As these z-points have a large distance from the center of the distribution they have a heavy impact on a PCA transformation to eigenvectors of the Pearson matrix. Therefore, I eliminated some outer points from the distribution. I did this by deleting points with coordinate values smaller or bigger than a symmetric threshold in the outer flanks of the 256 Gaussian distributions – e.g.:

z_j < mu_j – fact * FHWM and z_j > mu_j + fact * FHWM.

With FHWM meaning the full width at the half maximum and mu_j being the center of the approximate Gaussian for the j-th vector component of the z-vectors. I tried multiple values of fact between 1.7 and 1.25.

One has to be careful with such an elimination process. Even though we set our limits beyond the 2 σ-level of the individual Gaussians ahead of the transformation, a step-wise consecutive elimination for all components reduces the number of surviving vectors significantly for small values of fact. The plots below correspond to a value of fact = 1.7 (with one discussed exception). This reduced the number of available vectors from originally 170.000 used during the AE’s training to 160.000. From this distribution a random sub-selection of 80,000 z-points was taken to perform the PCA transformation. I call the resulting sample of z-points the “reduced distribution” below. The results did not change much for a sub-selection of more points. The second reduction saved me some CPU-time, however.

PCA transformation of the z-point distribution – explained variance and cumulative importance

Below you see the explained variance ratio of the first 40 and the cumulative importance of the first 120 main components after a PCA transformation of the reduced z-point distribution. (The PCA-algorithm was based on the SVD-algorithm, as provided by the sklearn-package.)

We see that we need around 120 PCA components to explain 80% of the variance of the original z-point data. The contribution of the first 15 components is significant, but it is NOT really dominant. Interesting is also the step-wise decline of the first 10 components in a not so smooth curve.

I have in addition checked that the normalized Pearson correlation coefficient matrix was perfectly diagonalized. All off-diagonal values were smaller than 2.e-7 (which is consistent with error propagation) and the diagonal elements were 1.0 with an accuracy better than 8 eight digits after the decimal point. As it should be.

Gaussians per PCA-vector component?

We expect Gaussian number density functions for each of the z-vector components after the PCA coordinate transformation. The following line plots show the number density distributions (colored line) and a related Gaussian fit (dashed lines) per named vector-components after PCA-transformation. The first plot gives an overview over 120 PCA components:

We see that all distributions look similar to Gaussians. In addition the center of the distributions for all components are very close to or coincide with the origin of the PCA coordinate system. This is something we would expect for an original multivariate normal distribution before the PCA transformation to a new moved and rotated coordinate system: An affine transformation between coordinate systems with orthogonal axes maps Gaussians to Gaussians.

Components 0 and 1:

Components 2 and 3:

Components 4 and 5:

Components 6 and 7:

Components 8 and 9:

We see from these plots that component 3 shows values at the distributions’s flanks well above the values of its Gaussian fit. This may lead to deformed ellipses. In addition component 7 shows a slight asymmetry – probably triggering deviations both at the center and outer regions. Similar effects can be seen for other higher components – but not so pronounced. The chosen sampling interval (0.5) in the new coordinates did smear out small asymmetric wiggles in all distributions.

Ellipses?

Calculating approximate contour lines is CPU intensive. On my presently available laptop I had to focus on some selected examples for pairs of components of z-vectors in the transformed coordinate system. Experiments showed that the most critical components with respect to asymmetries were 2, 3, 7, 8.

Let us first look at combinations of the first 9 PCA components (0) with other components in the range between the 10th and 120th PCA component. Just scroll down to see more images:

The fatter lines represent confidence ellipses derived both from correlation data, i.e. elements of the normalized correlation coefficient matrix, and properties of the Gaussian distributions. See the last blog for details on the method of getting confidence ellipses from data of a probability distribution.

Ellipses for density distributions in coordinate planes for pars of the most important PCA components

Before you think that everything is just perfect we should focus on the most important 10 PCA components and their mutual pair-wise density distributions. We expect trouble especially for combinations with the components 2, 3 and 7.

Component 0 vs other components < 10

We see already here that asymmetries in the density distributions which appear relatively small in the gaussian-like curves per component leave their imprint on the elliptic curves. The contour lines are not always centered with respect to the overall confidence ellipses.

Component 1 vs other components < 10

Component 2 vs other components < 10

The plot for components 2 and 3 was based on fact = 1.35 instead of fact = 1.7. See a section below for more information.

Component 3 vs other components < 10

Component 4 vs other components < 10

Component 5 vs other components < 10

Component 6 vs other components < 10

Component 7 vs other components < 10

Component 8 vs other components < 10

Component 9 vs other components < 10

Ellipses for other seected component pairs

Comments

The z-point distribution is rather close to a multivariate normal distribution, but it is not perfect. The PCA transformation revealed clearly that the center of the overall z-point distribution does not completely coincide with the centers of the individual density distributions for the z-vector components. And not all individual curves are fully symmetrical. Some plots show the expected ellipses, but not fully centered. We also see signs of non-linear correlations as not all ellipses show a complete alignment with the axes of the PCA coordinate system.

The impact of outer z-points becomes obvious when we compare plots for the component pair (2, 3) for different values of fact. The plots are for fact = 1.35, 1.45, 1.55, 1.7 and fact = 1.8 (in this order from left to right and downward). (Regarding z-point elimination the fact-values correspond to 122000, 138000, 149000, 159000 and 164000 remaining z-points.)

The flips in the left-right orientation between some of the plots can be ignored. They depend on random aspects of the transformations and plotting routines used.

We see that z-points (for some some strange images) which have a large distance from the center of the distribution have a major impact on the central density distribution. 10% of the outer points influence the orientation of the central ellipses by their coordinates – although these outer points are not part of the inner core of the distribution.

BUT: Overall we find the effect we wanted to see. The elliptic contours are clearly visible and most of them are aligned with the axes of the PCA coordinate system and the main axes of the confidence ellipses for the z-point distribution in the PCA coordinate system. The inner core is aligned with the axes of the coordinate system even for the critical PCA components 2 and 3, when we omit between 12% and 25% of the points (outside a 3 σ-level).

This means: The PCA transformation confirms that an inner core of the z-point distribution is well described by a multivariate normal distribution – with linear correlations between the number density distributions for various components.

This is an interesting finding by itself.

Conclusion

The last and the present post of this series have shown that a convolutional AE maps human face images of the CelebA data set to a multivariate normal distribution in its multidimensional latent space. Although this is interesting by itself we must not forget our ultimate goal – namely to generate random z-vectors which should deliver us reasonably human face images by the AE’s Decoder. But the results of our analysis provide solid criteria to generate such vectors fulfilling complex correlation conditions.

We can now define statistical generation methods which restrict the components of our aspired random z-vectors such that the resulting z-points lie within regions surrounded by confidence ellipses of the multivariate normal distribution. One such method is the topic of the next post.

Autoencoders and latent space fragmentation – X – a method to create suitable latent vectors for the generation of human face images

Links to other posts of this series

Autoencoders and latent space fragmentation – VIII – approximation of the latent vector distribution by a multivariate normal distribution and ellipses

Autoencoders and latent space fragmentation – VII – face images from statistical z-points within the latent space region of CelebA

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

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

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

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

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

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

 

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

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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

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

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))
fig.subplots_adjust(hspace=0.2, wspace=0.2)

#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" )
        axc.legend(labels=sp_names, handletextpad=0.1)
        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