Tag Archives: standard Autoencoder

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

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My present post series explores options to use a standard convolutional Autoencoder [AE] for the creation of images with human faces. The face generation should based on random input to the AE’s Decoder. On our quest for a suitable method we have meanwhile learned a lot about other aspects of Autoencoders, vector distributions in multi-dimensional latent spaces and generative methods for our special case:

  • Methods to create statistical latent vectors [z-vectors] as input for the AE’s Decoder must be chosen carefully. Among other things: It is difficult to create a bunch of random vectors which cover wider areas in the vastness of a multidimensional space. So the z-vector creation must be adjusted to specific requirements.
  • After having been trained with CelebA images a convolutional AE fills a limited and coherent region in the latent space with z-points for the training images. This latent space region appears to be critical for successful image creation: Statistically generated z-vectors should point to this region. The core of the z-point distribution gets filled relatively densely.
  • A convolutional AE maps human face images onto an approximate multivariate normal distribution. This gives the inner core of the z-point distribution the structure of a multidimensional ellipsoid. The projections of this ellipsoid onto 2-dimensional coordinate planes show characteristic nested elliptic contour lines.
  • As the main axes of these ellipses were inclined with different angle towards the axes of chosen coordinate planes we concluded that linear correlations mark average dependencies between the z-vector components. Limiting conditions imposed by these correlations must also be fulfilled by z-vectors used as the Decoder’s input.

See previous posts in this series for more details. In particular, the last 2 posts

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

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

have shown that the density distribution for the z-points really exhibits elliptic contour lines in the original coordinate system of the latent space and (!) in the target coordinate system of a PCA transformation.

In this post we use our gathered knowledge: I present a first simple method to generate z-vectors which point to the latent space region filled by z-points for CelebA images. These z-vectors will fulfill the general and limiting elliptic conditions for their components.

Decomposing the full problem of latent vector generation into a sequence of 2-dimensional problems

The nice thing about multivariate Gaussian distributions with linear correlations between the vector components is the following: We can reduce the problem of choosing proper component values to a series of 2-dimensional restrictions. Firstly we can use characteristic properties of the Gaussian distribution for each component. And secondly we can use confidence ellipses in 2-dimensional coordinate planes to restrict the component values to allowed intervals.

Ellipses are most easy to handle when their axes are aligned with the axes of the coordinate system in which we describe them. So, let us assume that we know an affine transformation T to a new coordinate system which also has orthogonal axes and supports the following special transformation properties for a multivariate normal density distribution:

  1. T maps nested elliptic contour lines of the multidimensional density distribution and in particular confidence ellipses for component pairs in the original coordinate system to nested elliptic contours and confidence ellipses in the new coordinate system.
  2. Taligns the centers of the transformed ellipses with the origin of the new coordinate system.
  3. T aligns the main axes of the mapped ellipses with the axes of the new coordinate system.
  4. T is reversible.

How could we then use the transformed data for vector-creation?

In the new coordinate system, a contour ellipse in a chosen coordinate plane for the axes-indices (i, j) may have main diameters of size

d1 = 2 * a    and    d2 = 2 * b.

We then can first select a random v_i value to fall into a range [-a * fact, a * fact].

– fact * a    <    v_i    <    fact * a

With fact being a proper factor. This factor defines a confidence level in the new coordinate system. With the value of v_i fixed and b being the half-diameter in the orthogonal direction the correlation condition for the z-point distribution says that the v_j value must fall into an interval [-c, c] defined by:

-c    <    v_j    <    c,
with c = b * fact * sqrt(1 – x**2 / (fact * a)**2)

But within these limits we can again choose the v_j-value freely. Below I use a simple random-function for a constant probability density to pick a value.

However: It would not be enough to restrict the coordinates to the conditions of just one ellipse! The components of the created vectors must in parallel fulfill elliptic conditions for all of the possible pairs of vector-components. I.e. we may need to adapt the v_j values gained from the analysis of a fist 2D-ellipse to further conditions of other ellipses and component pairs. This can be achieved by an iteration. For z_dim = 256 this involves a total of 32640 checks and possible value-adaptions to each and all of the allowed value ranges.

In addition: The order by which the component-pairs and their conditions are investigated must be randomized to get real statistical vector distributions.

Eventually the resulting vector components must be re-transformed into the original coordinate system of the latent space.

The ellipse for the “core’s boundary” in the original coordinate system will be defined by the chosen confidence level of the ellipsoidal normal distribution. We saw already that a confidence level of σ = 2.0 defines the transition to outer regions of the z-point density distribution quite well.

This all sounds manageable by relative simple Python programs. But: Do we know a proper transformation T? Yes, we do: A PCA-transformation of the z-point density distribution has all the properties discussed above.

Using half maximum values after a PCA transformation of the z-point distribution

The last post proved that a PCA transformation maps ellipses onto ellipses for component pairs in the transformed PCA coordinate system. The advantage of the ellipses there is that their main axes are on average well aligned with the orthogonal PCA coordinate axes. Gaussians for the number density distribution per component are mapped to Gaussians for the new components in the transformed coordinate system. So, the basic idea for a proper z-vector generation is:

  1. Take the multivariate normal z-point distribution for the training images in the AE’s latent space.
  2. Apply a PCA analysis to diagonalize the correlation matrix and transform the z-vector components to the PCA coordinate system.
  3. Use the ellipses in coordinate planes of the PCA coordinate system to create random z-vector components fulfilling all required conditions there.
  4. Re-transform the resulting z-vector components into the original coordinate system of the latent space.

Point 3 in our method is covered by a numerical analysis of the Gaussians in the PCA-coordinate system. We determine the half-width numerically by analyzing the density distribution with the help of sampling intervals. This simple method has resolution limits related to the size of the sampling interval. This has consequences for PCA components with a small standard deviation. We saw already in the last posts that such distributions appear for higher PCA components at the lower end of the explained variance.

Does the suggested method work?

The convolutional AE we work with was defined in previous posts with 4 Conv2D layers in the Encoder and 4 Conv2DTranspose layers in Decoder. The number of latent space dimensions was z_dim = 256. The AE network was trained on CelebA images. I do not want to bore you with details of the codes for the creation of z-vectors consistent to the resulting elliptic conditions. It is all standard. The PCA-transformation can e.g. be taken from the sklearn-package.

I have applied a constant probability density to choose a random value within the allowed ranges for the component values of the aspired z-vectors in the PCA coordinate system. For the plots below I have used the most important 50 to 105 PCA components (out of 256). The plots include confidence ellipses on a level of σ = 2.2. I derived the confidence ellipses by directly evaluating the standard deviations of the transformed distribution data in all coordinate directions.

The first plot shows you such an ellipse for the coordinate plane corresponding to the first two, most important PCA components. The orange points mark 20 z-points defined by 20 randomly z-vectors fulfilling all elliptic conditions. The plot contains 120,000 z-points for images out of the 170,000 CelebA pictures used during training.

Generated statistical vectors in the PCA coordinate system

For elliptic contour lines see the last post before the present one in this series. The next plot shows the same generated 20 z-vectors for other component-combinations among the first 20 of the most important PCA-components. The plots contain a selection of 60,000 z-points.

The outer z-points points do not always indicate that we have elliptic contours in the denser core of the displayed 2-dimensional distributions. But see the last post for proofs that the inner core inside the red ellipse really displays elliptic contours. You see that all random vectors lie within the 2-σ-ellipses.

The next plot shows the generated z-vectors in the original coordinate system of the latent space. The component values were back-transformed from the PCA-system to the original coordinate system.

Generated statistical z-vectors after an inverse PCA transformation to the original coordinate system of the latent space

We get similar plots for other component pairs. And of course for other generated vectors.

Generated statistical z-vectors in the PCA coordinate system

Generated statistical z-vectors after an inverse PCA transformation to the original coordinate system of the latent space

Technically we have obviously achieved what we wanted: Our generated statistical vectors are distributed within the core of our multidimensional ellipsoid.

Note that this method fortunately works even when we use a limited number of the PCA components, only. This is due to intricate properties of a PCA transformation which guarantee that a back-transformation puts the resulting points close to the original ones even when we omit less important PCA components. I cannot discuss the math-details in this blog. You have to see scientific literature for this. An introduction is e.g. provided by https://arxiv.org/pdf/1404.1100.pdf.

For me this property of the PCA transformation was helpful when I ran into the resolution problem for a proper half-width of the Gaussians. Taking 256 components lead to errors as elliptic conditions for very narrow Gaussians were not properly defined and some of the created vectors left the allowed value ranges.

Resulting face images

Let us look at some results. First I want to remind you from where we started:

Failed trials with improper random z-vectors based on constant probability densities

A simple random generator used in the beginning was totally inapt to feed the AE’s Decoder with proper statistical z-vectors. And now – look at the following plots. They were produced for a varying number of PCA components between 50 and 120, 100000 statistically selected z-points within a 3 σ-level for the PCA-transformation and various factors 0.6 < fact < 0.8 used upon a half-width corresponding to a confidence level of 2.35 σ:

In some cases – for a higher number of PCA components – we even see smaller details of the face images and a reasonable transition to some kind of hairdo. Please remember that z_dim = 256 is a pretty low number for the latent space to cover the encoding of face details. And celebrities as covered by CelebA use make-up ….

In case you think the above result is not noteworthy: Please remember that we talk about a simple standard Autoencoder and not about a Variational Autoencoder and neither about a transformer based Autoencoder. No fancy additions to cost functions or special layers. And who ever has read the very instructive book of D. Foster on “Generative Deep Learning” (1st edition, O’Reilly) may compare his images to mine. And I have used a lower resolution of the original images than D. Foster. Just to motivate people to look a bit deeper into properties of data distributions in latent spaces.

Conclusion and outlook

We have come a lot closer to our objective of using a standard minimal Autoencoder for generative purposes. On our way, we got a much deeper understanding of the vector-distribution a trained AE creates in its latent space for human face images.

The method presented in this post to create reasonable statistical z-vectors still has its limits and there is a lot of open space for improvements. Attentive readers may e.g. ask: Why did he not use confidence ellipses directly? And why not the ellipses found in the original coordinate system of the latent space? And what about micro-correlations? And are there clusters for certain properties as the hair-color, sex, smiling, etc. in the multivariate z-point distribution in the AE’s latent space?

I will discuss these topics in further posts. In the meantime keep in mind that the basic point for turning a standard Autoencoder into a generative tool is to understand how it fills its latent space.

Note also that I myself have speculated in other posts of this blog that failures of using standard AEs for generative purposes may have their ultimate reason in the micro-structure of the z-point distribution. The present results render these previous ideas of mine plain wrong.

Links to previous posts of this series

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

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

 

And before we forget it: Besides the Putler in the east there is also an extremist right-wing, semi-fascistic party in Germany on a record high support level in the population of 18%. This is a party which wants to stop all sanctions against the Russian aggressor in the ongoing war in Ukraine. You see the pattern behind this? This party is presently becoming bigger in number of supporters than the government leading social democrats. So, there is more at stake at present in Europe than the war in Ukraine. We need to defend our democracies with all the means of democracies. And its time to ask for more decisive legal action against a party which already is under observation of the German internal secret service.

 

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

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I continue with my analysis of the z-point and latent vector distribution a trained Autoencoder creates in its latent space for CelebA images. These images show human faces. To make the Autoencoder produce new face images from statistically generated latent vectors is a problem. See some previous posts in this series for reasons.

Autoencoders and latent space fragmentation – I – Encoder, Decoder, latent space
Autoencoders and latent space fragmentation – II – number distributions of latent vector components
Autoencoders and latent space fragmentation – III – correlations of latent vector components
Autoencoders and latent space fragmentation – IV – CelebA and statistical vector distributions in the surroundings of the latent space origin
Autoencoders and latent space fragmentation – V – reconstruction of human face images from simple statistical z-point-distributions?

These problems are critical for a generative usage of standard Autoencoders. Generative tasks in Machine Learning very often depend on a clear and understandable structure of the latent space regions an Encoder/Decoder pair uses. In general we would like to create statistical latent vectors such that a reasonable object creation (here: image creation) is guaranteed. In the last post

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

we saw that we at least get some clear face features when we make use of some basic information about the shape and location of the z-point distribution for the images the AE was trained with. This distribution is specific for an Autoencoder, the image set used and details of the training run. In our case the z-point distribution could be analyzed by rather simple methods after the training of an AE with CelebA images had been concluded. The number distribution curves per vector component revealed value limits per latent vector component. The core of the z-point distribution itself appeared to occupy a single and rather compact sub-volume inside the latent space. (The exact properties depend on the AE’s layer structure and the training run.) Of the N=256 dimensions of our latent space only a few determined the off-origin position of the center of the z-point distribution’s core. This multidimensional core had an overall ellipsoidal shape. We could see this both from the Gaussian like number distributions for the components and more directly from projections onto 2-dimensional coordinate planes. (We will have a closer look at these properties which indicate a multivariate normal distribution in forthcoming posts.)

As long as we kept the statistical values for artificial latent vector components within the value ranges set by the distribution’s core our chances that the AE’s Decoder produced images with new and clearly visible faces rose significantly. So far we have only used z-points along defined paths crossing the distributions core. In this post I will vary the components of our statistically created latent vectors a bit more freely. This will again show us that correlations of the vector components are important.

Constant probability for each component value within a component specific interval

In the first posts of this series I naively created statistical latent vectors from a common value range for the components. We saw this was an inadequate approach – both for general mathematical and for problem specific reasons. The following code snippets shows an approach which takes into account value ranges coming from the Gaussian-like distributions for the individual components of the latent vectors for CelebA. The arrays “ay_mu_comp” and “ay_mu_hw” have the following meaning:

  • ay_mu_comp: Component values of a latent vector pointing to the center of the CelebA related z-point distribution
  • ay_mu_hw: Half-width of the Gaussian like number distribution for the component specific values
num_per_row  = 7
num_rows     = 3
num_examples = num_per_row * num_rows

fact = 1.0

# Get component specific value ranges into an array 
li_b = []
for j in range(0, z_dim):  
    add_val = fact*abs(ay_mu_hw[j])
    b_l = ay_mu_comp[j] - add_val
    b_r = ay_mu_comp[j] + add_val
    li_b.append((b_l, b_r))
    
# Statistical latent vectors
ay_stat_zpts = np.zeros( (num_examples, z_dim), dtype=np.float32 )     
for i in range(0, num_examples): 
    for j in range(0, z_dim):
        b_l = li_b[j][0]
        b_r = li_b[j][1]
        val_c = np.random.uniform(b_l, b_r) 
        ay_stat_zpts[i, j] = val_c

# Prediction 
reco_img_stat = AE.decoder.predict(ay_stat_zpts)
# print("Shape of reco_img = ", reco_img_stat.shape)

The main difference is that we take random values from real value intervals defined per component. Within each interval we assume a constant probability density. The factor “fact” controls the width of the value interval we use. A small value covers the vicinity of the center of the CelebA z-point distribution; a larger fact leads to values at the border region of the z-point distribution.

Image results for different value ranges

fact=0.4

fact=0.5

fact=0.6

fact=0.7

fact=0.8

fact=0.9

fact=1.0

Selected individuals

Below you find some individual images created for a variety of statistical vectors. They are ordered by a growing distance from the center of the CelebA related z-point distribution.

Quality? Missing correlations?

The first thing we see is that we get problems for all factors fact. Some images are OK, but others show disturbances and the contrasts of the face against the background are not well defined – even for small factors fact. The reason is that our random selection ignores correlations between the components completely. But we know already that there are major correlations between certain vector components.

For larger values of fact the risk to place a generated latent vector outside the core of the CelebA z-point distribution gets bigger. Still some images interesting face variations.

Obviously, we have no control over the transitions from face to hair and from hair to background. Our suspicion is that micro-correlations of the latent vector components for CelebA images may encode the respective information. To understand this aspect we would have to investigate the vicinity of a z-point a bit more in detail.

Conclusion

We are able to create images with new human faces by using statistical latent vectors whose component values fall into component specific defined real value intervals. We can derive the limits of these value ranges from the real z-point distribution for CelebA images of a trained AE. But again we saw:

One should not ignore major correlations between the component values.

We have to take better care of this point in a future post when we perform a transformation of the coordinate system to align with the main axes of the z-point distribution. But there is another aspect which is interesting, too:

Micro-correlations between latent vector components may determine the transition from faces to complex hair and background-patterns.

We can understand such component dependencies when we assume that the superposition especially of small scale patterns a convolutional Decoder must arrange during image creation is a subtle balancing act. A first step to understand such micro-correlations better could be to have a closer look at the nearest CelebA z-point neighbors of an artificially created latent z-point. If they form some kind of pattern, then maybe we can change the components of our z-point a bit in the right direction?

Or do we have to deal with correlations on a much coarser level? What do the Gaussians and the roughly elliptic form of the core of the z-point distribution for CelebA images really imply? This is the topic of the next post.

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

 

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

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It is well known that standard (convolutional) Autoencoders [AEs] cause problems when you want to use them for creative purposes. An example: Creating images with human faces by feeding the Decoder of a suitably trained AE with random latent vectors does not work well. In this series of posts I want to identify the cause of this specific problem. Another objective is to circumvent some of the related obstacles and create reasonably clear images nevertheless. Note that I speak about standard Autoencoders, not Variational Autoencoders or transformer based Encoder/Decoder-systems. For basic concepts, terms and methods see the previous posts:

Autoencoders and latent space fragmentation – I – Encoder, Decoder, latent space
Autoencoders and latent space fragmentation – II – number distributions of latent vector components
Autoencoders and latent space fragmentation – III – correlations of latent vector components
Autoencoders and latent space fragmentation – IV – CelebA and statistical vector distributions in the surroundings of the latent space origin
Autoencoders and latent space fragmentation – V – reconstruction of human face images from simple statistical z-point-distributions?

So far I have demonstrated that randomly generated vectors most often do not hit the relevant regions in the AE’s latent space – if we do not take some data specific precautions. A relevant region is a confined volume which a trained Decoder fills with z-points for its training objects after the training has been completed. z-points and corresponding latent vectors are the result of an encoding process which maps digitized input objects into the latent space. Depending on the data objects we may get multiple relevant regions or just one compact region. In the case of a convolutional AE which I had trained with the CelebA dataset of human face images I found single region with a rather compact core.

In this post I want to create statistical latent vectors whose end-points are located inside the relevant region for CelebA images. Then I will create images from such latent vectors with the help of the AE’s Decoder. My hope is to get at least some images with clearly visible human faces. The basic idea behind this experiment is that the most important features of human faces are encoded by a few dominant vector components defining the overall position and shape of the multidimensional z-point region for CelebA images. We will see that the theory is indeed valid: Here is a first example for a vector pointing to an outer area of the core region for CelebA images in the latent space:

Our AE is a convolutional one. The number of latent space dimensions N was chosen to be N=256.
Note: We are NOT using a Variational Autoencoder, but a simple standard Autoencoder. The AE’s properties were discussed in previous posts.

What have we found out so far?

The Encoder of the convolutional AE, which I had trained with the CelebA dataset, mapped the human face images into a compact region of the latent space. The core of the created z-point distribution was located within or very close to a tiny hyper-volume of the latent space spanned by only a few coordinate axes. The confined multi-dimensional volume occupied by most of the z-points had an overall ellipsoidal shape with major extensions along a few main axes. We saw that some of the coordinates of the CelebA z-points and the components of the corresponding latent vectors were strongly correlated. In addition the value range of each of the latent vector components had specific individual limits – confining the angles and lengths of the vectors for CelebA. Therefore we had to conclude:

Whenever we base our method to create statistical vectors on the assumptions

  • that one can treat the vector components as independent statistical variables
  • that one can assign statistical values to the components from a common real value interval

the vectors will almost certainly not point to the relevant region. In addition one has to take into account unexpected mathematical properties of statistical vector distributions in high dimensional spaces. See the previous posts for more details. Indeed we could show that such a vector generation method missed the CelebA region.

Objective of this post

In this post I want to use some of the knowledge which we have gathered about the latent vector distribution for CelebA images. We shall use a very simple approach to probe the image reconstruction abilities of the Decoder for a defined variety of z-points:

We restrict the vectors’ component values such that most of the vectors point to the region formed by the bulk of CelebA z-points. To achieve this we define straight line segments which cross the ellipsoidal region of CelebA z-points. This is possible due to the known value intervals which we have identified for each of the components in a previous post. Then we place some artificial z-points onto our line segments. At least some of these z-points will fall into the relevant CelebA region. We then let the Decoder reconstruct images for the latent vectors corresponding to these z-points.

In some cases our paths will even respect some major component correlations, but for some paths I will explicitly disregard such correlations. Nevertheless our rather simple restrictions imposed on the vector-component values will already enable us to produce images with clearly recognizable face features.

Among other things our results confirm the idea that the real pixel correlations for basic face features are represented by relatively narrow limits for the angles and lengths of respective latent vectors. The extension and shape of the bulk region of CelebA z-points is defined by only a few latent vector components. These components apparently encode a prescription for the (convolutional) Decoder to create face features by a superposition of some elementary patterns extracted during the AE’s training.

A path from the latent space origin to the center of the relevant z-point region

How do we restrict latent vectors to the required value ranges? In the 2nd post we have seen that the number distribution curve for the values of each of the latent vector components was very similar to a Gaussian. We have identified the mean value and average value range for each component by analyzing its specific distribution curve. The mean values gave us the coordinates of the center of the relevant latent space region. In addition we, of course, know the coordinates of the origin of the latent space. So, for a first test, let us create a multi-dimensional line segment between the origin and the center of the CelebA z-point distribution. And let the A’s Decoder create images for latent vectors pointing to some intermediate z-points along this path.

The following plots show orthogonal projections of 5000 CelebA z-points (in blue) onto some 2-dimensional planes spanned by two selected coordinate axes. The yellow dot indicates the origin. The orange dot the center of the z-point distribution. Red dots indicate coordinates of points along the straight path between the origin and the distribution center.

Please, take note of the different scales on the x- and y-axes. Some distributions are much more elongated than the scaled images show. That some paths appear shorter than others is due to the projection of the diagonal line through the multi-dimensional space onto planes which are differently oriented with respect to this line. A simple 3D analog should make this clear. Some small wiggles in the positions of the red dots are due to resolution problems of the plot on the browser interface. We also see a reflection of the fact that the origin is located in a border region of the bulk.

Below you see a plot which shows the path in higher resolution (projected onto a particular plane):

Again: Take note of the different axis scales. The blue dot distribution is much more stretched in C1-direction than it appears in the plot.

Ok, now we have a multidimensional path and six well defined latent vectors for the end and intermediate points on this path. So let us provide these vectors as input to the our AE’s Decoder. The resulting images look like:

Success! Images in the surroundings of the center show a clearly visible face. And we also see: The average face at the center of the z-point distribution is female – at least according to the CelebA dataset. 🙂 However: In the vicinity of the origin of the latent space we get no images with reasonable face features.

Images along a path within a selected coordinate plane for two dominant vector components

I choose a different path within the plane spanned by the coordinates axes 151 and 195 now. This is depicted in the plot below:

A look into the second post shows you that the components 151, 195 were members of the group of dominant components. Those were components for which the number distribution showed a mean value at some distance from the origin of the latent space and also had a half-width bigger than 1.0 (as most of the other components). The images reconstructed by the Decoder from the latent vectors are:

Hey, we get some variation – as expected. Now, let us rotate the path in the plane:

Not so much of a difference. But we have learned that a variation of some vector component values within the allowed range of values may give us already some major variation in the faces’ expressions.

Images for other coordinate planes

The following images show the variations for paths in other coordinate planes. All of the paths have in common that they pass the center of the CelebA bulk region. For the first 4 examples I have kept the path within the core region of CelebA z-points. The last images show images for paths with z-points at the core’s border regions or a bit outside of it.

Plane axes: 5, 8

Plane axes: 17, 180

Plane axes: 44, 111

Plane axes: 55, 56

Plane axes: 15, 242

Plane axes: 58 202

Plane axes: 68, 178

Plane axes: 177, 202

Plane axes: 180, 242

The images for z-points farther away from the bulk’s center give you more interesting variations. But obviously in the outer areas of the CelebA region correlations between the latent vector components get more important when we want to avoid irregular and unrealistic disturbances. All in all we also get the impression that a much more subtle correlation of component values is a key for the reproduction of realistic transitions for the hairdos presented in the CelebA images and the transition to some realistic background patterns. The components of our latent vectors are still too uncorrelated for such details and an appropriate superposition of micro-patterns in the images created by the Decoder.

Conclusion

This blog shows that we do not need a Variational Autoencoder to produce images with recognizable human faces from statistical latent vectors. We can get image reproductions with varying face features also from the Decoder of a standard convolutional Autoencoder. A basic requirement seems to be that we keep the vector components within reasonable value intervals. The valid component specific value ranges are defined by the shape of the compact hyper-volume, which an AE’s Encoder fills with z-points for its training objects. So we need to construct statistical latent vectors which point to this specific sub-region of the latent space. Vectors with arbitrary components will almost certainly miss this region and give no interpretable image content.

In this post we have looked at vectors defining z-points along specific line segments in the latent space. Some of the paths were explicitly kept within the inner core regions of the z-point-distribution for CelebA images. From these z-points the most important face features were clearly reconstructed. But we also saw that some micro-correlations of the latent vector components seem to control the appearance of the background and the transition from the face to hair and from the hair to the background-environment.

I have not yet looked at line segments which do not cross the center of the bulk of the z-point distribution for CelebA images in the latent space. But in the next post

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

I first want to look at z-points for which we relatively freely vary the component values within ranges given by the respective number distributions.