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

Posted on 24. May 2023 by eremo

This post series is about creative abilities of convolutional Autoencoders [AE] which have been trained on a set of human face images. The objectives of this series and its numerical experiments are:

We want to create images with human faces from statistical z-vectors and related z-points in the AE’s latent space [z-space or LS]. Image creation will be done with the help of the AE’s Decoder after a training on the CelebA dataset.
We work with a standard Autoencoder, only. I.e., we do NOT add any artificial layers and cost terms to the Autoencoder’s layer structure (as it is done e.g. in Variational Autoencoders).
We analyze the position, shape and internal structure of the multidimensional z-vector distribution created by the AE’s Encoder after training. We assume that generated statistical z-vectors must point to respective regions of the latent space to guarantee images with reasonable content.
We raise the question whether simple statistical generator algorithms are sufficient to cover these regions with statistical z-vectors.

Our numerical experiments gave us some indications that such an endeavor is indeed feasible. In addition the third objective may give us some insight into the rules a trained AE follows when it encodes information about human faces into vectors of its latent space.

We have already studied the “natural” z-vector distribution created by a convolutional Autoencoder for CelebA images after a thorough training. The related z-point distribution fortunately filled just one confined and coherent off-center region of the AE’s latent space. Our experiments have furthermore shown that we must indeed restrict the statistical z-vector creation such that the vectors point to this particular region. Otherwise we will not get reasonable images. For details see the previous posts.

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

The frustrating point so far was that simple methods for creating statistical vectors fail to put the end-points of the z-vectors into the relevant latent space region. In particular methods based on constant probability distributions within a common value interval for all z-vector components are doomed to miss the interesting region due to intricate mathematical reasons.

Afterward we tried to restrict the component values of test vectors to intervals defined by the shape of the number distribution for the values of each component of the CelebA related z-vectors. Such a distribution is nothing else than a one-dimensional probability density function for our special set of encoded CelebA samples: The function describes the probability that a component of a z-vector for human face images gets a value within a certain small value range. The probability distributions for all z-vector components were bell shaped and showed clear transitions to flat wings with very low values. See the plots below. This allowed us to define a value range

d_j_l < x_j < d_j_h

for each vector component x_j.

But keeping statistical values per component within the identified respective interval was not a sufficient restriction. We saw this clearly in the last post from significant irregular fluctuations in the reconstructed images. Obviously the components of statistically generated z-vectors must in addition fulfill correlation conditions.

The questions which I want to answer in this post are:

Can we approximate the 1-dimensional probability density functions for the z-vector components by some simple and common mathematical function?
What kind of correlations do we find between the components of the z-vectors encoding the information of human face images?
Can we derive some mathematical description of the multivariate z-vector distribution created by convolutional AEs for human face images in the AE’s multidimensional latent space?

Correlations are to be expected …

Please note: We deal with a multidimensional problem. A single latent vector encodes information about a human face image via all of its component values and by relations between these values. Regarding the purpose and the task an AE has to fulfill, it would be naive to assume that the components of our multi-dimensional z-vectors were independently organized. A z-vector encodes information for a convolutional Decoder to combine patterns detected by the Encoder and represented in neural (feature) maps of the networks to create an image. This is a subtle business. Just think about what you do when you draw a sketch of a human face. There are a lot of rules you follow.

When you think about the properties of basic feature patterns in a human face you would certainly assume that the pixel data of a corresponding image show strong correlations. This is among other things due to obvious symmetries – not excluding fluctuations of basic parameters describing human face features. But a nose tends to be at a position below the eyes and at a mid-distance of the eyes. In additions fluctuations of face features would on average respect certain limits given by natural proportions of a face. It would therefore be unreasonable to assume that the input for a Decoder to create a superposition of elementary patterns consists of un-correlated data. Instead the patterns in the original data should not only lead to well adjusted weights in the convolutional networks’ feature maps, but also to well regulated structural elements in the data distribution in the target space of the information encoding, namely in the latent space.

If the relations of the vector components were of a complex, highly non-linear kind and involved many dimensions at the same time we might be lost. But the results we have gained so far indicate a proper common structure of at least the density function for the individual components. This gives us some hope that the multidimensional problem somehow involves well defined 1-dimensional constituents. Whether this a sign that the multidimensional structure of the z-vector distribution can be decomposed into low-dimensional relations remains to be seen.

Observations regarding the z-vector distribution created by convolutional Autoencoders for human face images

Coordinate values of the z-points are identical to z-vector component values when we fix the end of each vector to the origin of the latent space coordinate system. The z-vector distribution thus directly corresponds to a z-point density distribution in the orthogonal coordinate system of the AE’s multi-dimensional LS. We have already made three interesting observations regarding these distributions:

The individual probability density function for a selected component of the latent vectors has a bell-shaped form. One , therefore, is tempted to think of a Gaussian function. This would indicate a possible normal distribution for the coordinate values of the z-points along each of the selected coordinate axes.
Note: This does not exclude that the probability distributions for the components are correlated in some complex way.
When we plotted the projection of the z-point distribution onto 2-dimensional coordinate planes (for selected pairs of coordinate axes) then almost all of the resulting 2-dimensional density distributions seemed to have a defined core with an ellipsoidal form of its boundary.
For certain component- or axis-pairs the main axes of the apparent ellipses for pair-wise density function appeared rotated against the coordinate axes. The elongated regular and more or less symmetric forms showed a diagonal orientation (with different angles). This alone signals a strong correlation between related two vector components. Indeed we found high values for certain elements of the matrix of normalized Pearson correlation coefficients for the multi-dimensional distribution of z-vector component values.

These observations are not unrelated; they indicate a clear pattern of dependencies and correlations of the distributions for the variables in place. Regarding the data basis we have to keep five things in mind:

We treat the z-point distribution for CelebA images as a multi-dimensional probability density distribution. During the analysis we look in particular at 2-dimensional projections of this distribution onto planes spanned by a selected pair of orthogonal axes of the LS coordinate system. We also consider the one-dimensional value distributions for z-vector components. In this sense we regard the z-vector components as logically separate variables.
The data used are numbers of z-points counted in finite 1d-intervals, 2d-rectangles or multidimensional cuboids. We fit idealized functions to the respective discrete bar plots. Even if there is a good 1d-fit fluctuations may especially get visible in multidimensional plots for correlated data. A related probability density requires a normalization. We drop the resulting constant factors in the qualitative discussions below.
Statistical (un-)correlation of statistical variable distributions must NOT to be confused with underlying variable (in-) dependency. Linear correlations can be reduced to zero by coordinate transformations without eliminating the original variable dependencies.
Pearson correlation coefficients are sensitive to linear elements in the relations of logically separate variable distributions. They can not fully cover non-linear distribution relations or covered variable dependencies.
A transformation to a local coordinate system whose axes are aligned to the so called main axes of the multidimensional distributions does not remove the original data relations – but there may exist a coordinate system in which the distribution data can be described in a simple, factorized form corresponding to a composition of seemingly un-correlated data distributions.

Anyway – by discussing density distributions we work on overall and large scale average relations between statistical value distributions for our variables, namely the z-vector components. We do not cover local micro-relations that may be in place in addition.

The relation of ellipses with Gaussian probability densities

Probability density functions for two logically separate, but maybe not un-correlated variables have to be multiplied. In our case this reflects the following point: First we determine the probability that the value of component x_i lies in a certain (infinitesimal) interval and then we determine the probability that (for the given value of x_i) the component x_j falls into another value range. The distributions for a specific variable can include variable relations and thus the probability density g(x_j) can include a dependency g(x_j(x_i)).

In the case of uncorrelated normal distributions per coordinate we can just multiply the individual Gaussians g(x_i) * g(x_j). Due to the quadratic terms in the exponent of the Gaussians we then get a sum of quadratic expressions in the common exponent, having the form fac1 * (x_i-mu_i)**2 + fac2 * (x_j-mu_j)**2.

By setting this expression to a constant value we get contour lines of the probability density distribution for the (x_i, x_j)-distribution. Quadratic sums correspond to the definition of an ellipse having main axes which are aligned with the x_i- and x_j-axes of the coordinate system. Thus the contour lines of a 2-dimensional distribution composed of un-correlated Gaussians are ellipses having an orientation aligned with the coordinate axes.

This was for un-correlated density-distributions of two vector components. Mathematically a linear correlation between a pair of Gaussians-distributions corresponds to an affine transformation of the contour-ellipses. The transformation can be expressed by a defined sequence of matrix operations describing a translation, rotations (in a defined order) and a dilation.

This means: The contour lines for a 2-dimensional probability density composed of linearly correlated Gaussians are still ellipses. But these ellipses will appear to be shifted, rotated and stretched along the main axes in comparison with their originally un-correlated Gaussian counterparts. The angle of rotation depends on details of the correlation function and the original standard deviations. The Pearson correlation matrix for linearly correlated distributions is a positive-definite one and, of course, shows off-diagonal elements different from zero. This result can be extended to multivariate normal distributions in spaces with many dimensions and related affine transformations of the coordinate system.

A multivariate normal distribution with linear correlations between the Gaussians results in elliptic contour lines for pair-wise density distributions in the respective 2D-coordinate planes of an orthogonal coordinate system. When we define the contours via multiples of the standard deviations of the underlying Gaussian functions we arrive at so called confidence ellipses.

A really nice mathematical aspect is that the basic parameters of the confidence ellipses can be derived from the normalized correlation coefficients of the Pearson matrix of the multivariate probability distribution. I will come back to this point in forthcoming posts in more detail. For now we just need to know that a multidimensional probability density comes along with confidence ellipses which can be calculated with the help of Pearson correlation coefficients.

Before we go on a word of caution: For a general multi-variate distribution it is not at all clear that it should decompose into a factorized form. However, for a multivariate normal distribution with un-correlated or only linearly correlated components this is by definition different. In this case a transformation to a coordinate system can be found which leads to a complete decomposition into a product of (seemingly) un-correlated Gaussians per component. The latter point lies at the center of PCA and SVD algorithms, which diagonalize the Pearson correlation matrix.

Do we really have Gaussian probability distributions for the individual z-vector-components?

After this short tour into the world of (multi-variate) normal distributions, Gaussian functions and related ellipses we are a bit better equipped to understand the density distributions in the latent space of our Autoencoders for human face images.

Let me remind you about the shapes of the number distribution for our concrete z-vector components resulting for for CelebA face images. The first plot shows the number densities on sampling intervals of width 0.25 for selected vector components resulting for case I of our experiments. The second plot shows the number densities for the values of selected components of case II.

Ok, these curves do resemble Gaussians and some fluctuations are normal. But can we prove the Gaussian properties of the curves a bit better?

Well, for case II I have drawn the best fits by Gaussian functions with the help of SciPy’s optimize.curve_fit() for 3 and yet another 4 selected components of the latent vectors and the respective number distribution curves. The dashed lines show the approximations by Gaussian functions:

The selected components are part of the list of around 20 dominant component distributions – due to their relatively large standard deviations. But the Gaussian form is consistently found for all components (with some small deviations regarding the symmetry of the curves).

So all in all it looks like as if our convolutional AE has indeed created a multivariate normal z-point distribution in the latent space. As said: This does not exclude correlations …

Pairwise linear correlations of the (normal) probability distributions for the latent vector components?

Now we are a bit bold – and assume the best case for us: Could the approxiate Gaussians distributions for the component values be pair-wise and linearly correlated? What would be a clear indication of a pair-wise linear correlation of our component distributions?

Well, we should find an elliptic form of contour lines in the 2-dimensional distribution for the component pair in the respective coordinate plane of the basic orthogonal LS coordinate system. This imposes quite strong symmetry conditions on the contour lines. The ellipses can be shifted and rotated – but they should remain being ellipses. If non-linear contributions to the correlation had a significant impact this would not be the case.

Practically it is not trivial to prove that we have approximately rotated ellipses in 2 dimensions. Satter plos alone do not help: Ellipses fit a lot of plotted distributions of discrete data points quite well. We really need to count number densities to get reliable contour lines. The following plots show such contour lines based on number sampling in rectangles and local smoothing operations with the help of scipy.stats.gaussian_kde().

The fat red and dark orange lines show corresponding confidence ellipses derived from the original CelebA distribution. See below for some remarks on confidence ellipses.

The contours are basically of elliptic shape although they do not show the complete symmetry expected for pure and linearly dependent Gaussian distributions. But overall the confidence ellipses fit quite well into the general form and orientation of the distributions. We also see that for higher σ-levels the coincidence with nearby contours is quite good. The wiggles in the contour change with the z-vector selection a bit.

We conclude that our basic impression regarding an elliptic shape of the z-point distributions is basically consistent with only linearly correlated Gaussian probability density distributions for the component values of the latent vectors.

Approximation of the core of the multivariate z-point distribution by confidence ellipses for component pairs

Above I referred to the boundary of a core of the probability density for two selected vector components. But how would we define the “boundary” of a continuous distribution in the coordinate planes? Answer: As we like – but based on the decline of the approximate Gaussian curves.

We can e.g. pick two times the half-width in each direction or we can use contours defined by confidence levels.
For 2 ≤ fact * σ ≤ 3 we saw already that the contour lines could well be fitted by confidence ellipses. A 3-sigma level covers around 97% of all data points or more. A 2 sigma-level ellipse encircles between 70% and 90% of all data points, depending on the eccentricity of the ellipse. Note that the numbers are smaller for ellipses than for rectangles. I.e. the standard 68-95-99.7 rule does not apply.

The plots below give you an impression of how well ellipses for a 2σ-confidence level approximate the core of the CelebA distribution in selected 2D coordinate planes of the latent space:

Each of the sub-plots was based on 10,000 statistically selected vectors of the 170,000 available in my test runs. This is a relative low number. Therefore, for a certain diameter of the points in the scatter plot only the inner core appears to be densely populated. The next plots shows the results for a 3 σ-level of the ellipse – but this time for 50,000 vectors. With more vectors we could visually fill the outer regions of the core.

The orange points mark the center of the multidimensional distributions derived from the one-dimensional distribution curves for the components. We see that it does not always appear to be optimally centered. There are multiple reasons: Our functions are not fully symmetric as ideal Gaussians. And equally important: The accuracy of the position depends on the sampling resolution which was coarse. Outliers of the distribution do have an impact.

And how would we explain the appearance of Gaussians and ellipses?

This all looks quite good, despite some notable deviations regarding symmetry and maxima. Gaussians fit at least most of the important probability density curves very well, though not by a 100%. The appearance of an elliptic shape of the inner core of the distribution and the appearance of overall elliptic contour curves can be explained by linear correlations of the Gaussian distributions for the components.

The appearance of normal distributions per component and basically linear correlations is something that really should be explained. I mean, dwell a bit on what we have found:

A convolutional Autoencoder network with more than 10 million adjustable parameters encoded information about human face images in the form of a roughly multivariate normal distribution of z-points in its latent space – with basically linear correlations between the Gaussian curves describing the probability densities functions for the component values of the z-vectors.

I find this astonishing and not at all self-evident. It is one of the most simple solutions for a multidimensional situation one can imagine. The following questions automatically came to my mind:

Does such a result only appear for training images of defined objects with some Gaussian variation in their features? Are the normal distributions a reflection of variations of relevant features in the original data?
Is this a typical result for (convolutional) AEs? How does it depend on the dimensionality of the latent space? Does it automatically come with a large number of z-space dimensions? Is it an efficient way to encode feature differences in the latent space, which (convolutional) AEs in general tend to use due to their structure?

Do I personally have a convincing explanation? No. Especially not, as the data shown above stem from convolutional neural networks [CNNs] without any batch-normalization layers.

A first idea would be that the dominant features of a human face themselves show variations described by Gaussian normal distributions already in the original data and that convolutional filtering does not destroy such distributions during optimization. A problem of this idea lies in the (non-) linear activation functions used at the nodes of the neural maps. Though ReLU, Leaky ReLU and SeLU contain linear parts.

The other problem is the linear form of the correlations. This is a rather simple kind of correlations. But why should an AE choose this simple form into its mapping of image information to latent space vectors after training?

How to generate statistical vectors for the creation of human face images?

The positive message which comes with the above results is that our problem of how to create proper statistical z-vectors decomposes into a sequence of two-dimensional problems. We can use the data of the ellipses appearing in the density-distributions for pairs of vector components to confine the components of statistically generated z-vectors to the relevant region in the latent space. All ellipses together restrict the component values in a well defined form. In the next post I will shortly outline some methods of how we can use the information contained in the ellipses with available algorithms.

Conclusion

In this post we have seen that for the case of a convolutional Autoencoder trained on CelebA human face images the latent vector distributions showed some remarkable properties:

The probability density functions for all component values can roughly be approximated by Gaussian functions. The components appear to be pairwise linearly correlated – at least to first order analysis. This automatically implies elliptic contour curves for the pairwise number density functions of coordinate values. Such contour curves were indeed found with first order accuracy. The core of the probability density for the z-points in the latent space could therefore be approximated by confidence ellipses for a σ-level above σ = 2.5.
The elliptic conditions correspond to a multivariate normal distribution with linear correlations of the variables.

Before we get to enthusiastic about these findings we should be careful and await a further test. All statements refer to a first order approximations. A real multivariate normal distribution would decompose into un-correlated Gaussians and 2D-ellipses of probability densities of component pairs after a PCA transformation.

In the next post

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

I shall present the results of a PCA analysis. In later posts I will introduce a related method to restrict the components of statistical vectors to the relevant region in the latent space of our Autoencoder.

Links and literature

On first sight my short description of the relation between multivariate Gaussian normal distributions and ellipses as the contour lines for the projected density distributions on coordinate planes may appear plausible. But in the general multi-dimensional case the question of linear correlations requires some more math than indicated. For details I just refer to some articles on the Internet – but any good book on multivariate analysis will give you the relevant information
https://de.wikipedia.org/ wiki/ Multivariate_ Normalverteilung
https://de.wikipedia.org/ wiki/ Mehrdimensionale_ Normalverteilung
http:// www.mi.uni-koeln.de/ ~jeisenbe/ Vortrag2.pdf
https://methodenlehre.uni-mainz.de/ files/ 2019/06/ Multivariate-Distanz-Normalverteilung-MDC-Bayes.pdf
https://en.wikipedia.org/ wiki/ Multivariate_ normal_ distribution
https://en.wikipedia.org/ wiki/ Confidence_region
https://users.cs.utah.edu/ ~tch/ CS6640F2020/ resources/ How to draw a covariance error ellipse.pdf
https://biotoolbox.binghamton.edu/ Multivariate Methods/ Multivariate Tools and Background/ pdf files/ MTB%20070.pdf

Regarding the intimate relation between the ellipses’ main axes to normalized Pearson correlation coefficients I also refer to
https://carstenschelp.github.io/ 2018/09/14/ Plot_ Confidence_ Ellipse_ 001.html
I am very grateful that the author Carsten Schelp saved me a lot of time when trying to find a way to program a solution for confidence ellipses. Thank you, Mr. Schelp for the great work.

Google Colab, RAM, VRAM and GPU usage limits – I – no clear conditions over multiple sessions

Posted on 4. May 2023 by eremo

I am a retired physicist with a hobby: Machine Learning [ML]. I travel sometimes. I would like to work with my ML programs even when I only have a laptop available, with inadequate hardware. One of my ex-colleagues recommended Google Colab as a solution for my problem. Well, I am no friend of the tech giants and for all they offer as “free” Cloud services you actually pay a lot by giving them your personal data in the first place. My general experience is also that you sooner or later have to pay for resources a serious project requires. I.e. when you want and need more than just a playground.

Nevertheless, I gave Colab a try some days ago. My first impression of the alternative “Paperspace” was unfortunately not a good one. “No free GPU resources” is not a good advertisement for a first time visitor. When I afterward tried Google’s Colab I directly got a Virtual Machine [VM] providing a Jupyter environment and an optional connection to a GPU with a reasonable amount of VRAM. So, is everything nice with Google Colab? My answer is: Not really.

Google’s free Colab VMs have hard limits regarding RAM and VRAM. In addition there are unclear limits regarding CPU/GPU usage over multiple sessions in an unknown period of days. In this post series I first discuss some of these limits. In a second post I describe a few general measures on the coding side of ML projects which may help to make your ML project compatible with RAM and VRAM limitations.

The 12.7 GB RAM limit for the RAM of free Colab VMs

Even for mid-size datasets you soon feel the 12.7 GB limit on RAM as a serious obstacle. Some RAM (around 0.9 to 1.4 GB) is already consumed by the VM for general purposes. So, we are left with around 11 GB. My opinion: This is not enough for mid-size projects with either big amounts of text or hundreds of thousands of images – or both.

When I read about Colab I found articles on the Internet saying that 25 GB RAM was freely available. The trick was to drive the VM into a crash by an allocation of too much RAM. Afterward Google would generously offer you more RAM. Really? Nope! This does not work any more since July 2020. Read through the discussion here:

Google instead wants you to pay for Google Pro. But as reports on the Internet will tell you: You still get only 25 GB RAM with Pro. So as soon as you want to do some serious work with Colab you are supposed to pay – a lot for Colab Pro+. This is what many professional people will do – as it often takes more time to rework the code than just paying a limited amount per month. I shall go a different way ..

Why is RAM consumption not always negative?

I admit: When I work with ML experiments on my private PCs, RAM seldom is a resource I think about much. I have enough RAM (128 GB) on one of my Linux machines for most of the things I am interested in. So, when I started with Colab I naively copied and ran cells from one of my existing Jupyter notebooks without much consideration. And pretty soon I crashed the VMs due to an exhaustion of RAM.

Well, normally we do not use RAM to a maximum for fun or to irritate Google. The basic idea of having the objects of a ML dataset in a Numpy array or tensor in RAM is a fast transfer of batch junks to and from the GPU – you do not want to have a disk involved when you do the real number-crunching. Especially not for training runs of a neural network. But the limit on Colab VMs make a different and more time consuming strategy obligatory. I discuss elements of such a strategy in the next post.

15 GB of GPU VRAM

The GPU offer is OK from my perspective. The GPU is not the fastest available. However, 15 GB is something you can do a lot with. Still there are data sets, for which you may have to implement a batch based data-flow to the GPU via a Keras/TF2 generator. I discuss also this approach in more detail in the next post.

Sometimes: No access to a GPU or TPU

Whilst preparing this article I was “punished” by Google for my Colab usage during the last 3 days. My test notebook was not allowed to connect to a GPU any more – instead I was asked to pay for Colab Pro. Actually, this happened after some successful measures to keep RAM and VRAM consumption rather low during some “longer” test runs the day before. Two hours later – and after having worked on the VMs CPU only – I got access to a GPU again. By what criterion? Well, you have no control or a clear overview over usage limits and how close you have come to such a limit (see below). And uncontrollable phases during which Google may deny you access to a GPU or TPU are no conditions you want to see in a serious project.

No clear resource consumption status over multiple sessions and no overview over general limitations

Colab provides an overview over RAM, GPU VRAM and disk space consumption during a running session. That’s it.

On a web page about Colab resource limitations you find the following statement (05/04/2023): “Colab is able to provide resources free of charge in part by having dynamic usage limits that sometimes fluctuate, and by not providing guaranteed or unlimited resources. This means that overall usage limits as well as idle timeout periods, maximum VM lifetime, GPU types available, and other factors vary over time. Colab does not publish these limits, in part because they can (and sometimes do) vary quickly. You can relax Colab’s usage limits by purchasing one of our paid plans here. These plans have similar dynamics in that resource availability may change over time.”

In short: Colab users get no complete information and have no control about resource access – independent of whether they pay or not. Not good. And there are no price plans for students or elderly people. We understand: In the mindset of Google’s management serious ML is something for the rich.

The positive side of RAM limitations

Well, I am retired and have no time pressure in ML projects. For me the positive side of limited resources is that you really have to care about splitting project processes into cycles for scaleable batches of objects. In addition one must take care of Python’s garbage collection to free as much RAM as possible after each cycle. Which is a good side-effect of Colab as it teaches you to meet future resource limits on other systems.

My test case

As you see from other posts in this blog I presently work with (Variational) Autoencoders and study data distributions in latent spaces. One of my favorite datasets is CelebA. When I load all of my prepared 170,000 training images into a Numpy array on my Linux PC more than 20 GB RAM are used. (And I use already centered and cut images of a 96×96 pixel resolution). This will not work on Colab. Instead we have to work with much smaller batches of images and work consecutively. From my image arrays I normally take slices and provide them to my GPU for training or prediction. The tool is a generator. This should work on Colab, too.

One of my neural layer models for experiments with CelebA is a standard Convolutional Autoencoder (with additional Batch Normalization layers). The model was set up with the help of Keras for Tensorflow 2.

First steps with Colab – and some hints

The first thing to learn with Colab is that you can attach your Google MyDrive (coming with a Google account) to the VM environment where you run your Jupyter notebooks. But you should not interactively work with data files and data sets on the mounted disk (on /content/MyDrive on the VM). The mount is done over a network and not via a local system bus. Actually transfers to MyDrive are pretty slow – actually slower than what I have experienced with sshfs-based mounts on other hosted servers. So: Copy singular files to and from MyDrive, but work with such files on some directory on the VM (e.g. under /home) afterward.

This means: The first thing you have to take care of in a Colab project is the coding of a preparation process which copies your ML datasets, your own modules for details of your (Keras) based ML model architecture, ML model weights and maybe latent space data from your MyDrive to the VM.

A second thing which you may have to do is to install some helpful Python modules which the standard Colab environment may not contain. One of these routines is the Nvidia smi version for Python. It took me a while to find out that the right smi-module for present Python 3 versions is “nvidia-ml-py3”. So the required Jupyter cell command is:

!pip install nvidia-ml-py3

Other modules (e.g. seaborne) work with their standard names.

Conclusion

Google Colab offers you a free Jupyter based ML environment. However, you have no guarantee that you always can access a GPU or a TPU. In general the usage conditions over multiple sessions are not clear. This alone, in my opinion, disqualifies the free Colab VMs as an environment for serious ML projects. But if you have no money for adequate machines it is at least good for development and limited tests. Or for learning purposes.

In addition the 12 GB limit on RAM usage is a problem when you deal with reasonably large data sets. This makes it necessary to split the work with such data sets into multiple steps based on batches. One also has to code such that Python’s garbage collection can work on small time periods. In the next post I present and discuss some simple measures to control the RAM and VRAM consumption. It was a bit surprising for me that one sometimes has to manually care about the Keras Backend status to keep the RAM consumption low.

Links

Tricks and tests
https://damor.dev/your-session-crashed-after-using-all-available-ram-google-colab/
https://github.com/ googlecolab/ colabtools/ issues/253
https:// www.analyticsvidhya.com/ blog/ 2021/05/10-colab-tips-and-hacks-for-efficient-use-of-it/

Alternatives to Google Colab
See a Youtube video of an Indian guy who calls himself “1littlecoder” and discusses three alternatives to Colab: https:// www.youtube.com/ watch?v=xfzayexeUss

Kaggle (which is also Google)
https://towardsdatascience.com/ kaggle-vs-colab-faceoff-which-free-gpu-provider-is-tops-d4f0cd625029

Criticism of Colab
https://analyticsindiamag.com/ explained-5-drawback-of-google-colab/
https://www.reddit.com/ r/ GoogleColab/ comments/ r7zq3r/ is_it_just_me_ or_has_google_colab_ suddenly_gotten/
https://www.reddit.com/ r/ GoogleColab/ comments/ lgz04a/ regarding_ usage_limits_ in_colab_ some_common_sense/
https://github.com/ googlecolab/ colabtools/ issues/1964
https://medium.com/ codex/ can-you-use-google-colab-free-version-for-professional-work-69b2ba4392d2

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

Posted on 2. May 2023 by eremo

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

Posted on 30. April 2023 by eremo

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:

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.

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

Posted on 24. April 2023 by eremo

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

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

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

that we can treat the component values of latent vectors as independent variables
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.