# Latent spaces – pitfalls of distributing points in multi dimensions – II – missing specific regions

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In this post series I discuss results of a private study about some simple statistical vector distributions in multi-dimensional latent vector spaces. Latent spaces often appear in Machine Learning contexts and can be represented by the ℜN. My main interest is:

What kind of regions of such a space may we miss by choosing a vector distribution based on a simple statistical creation process?

This problem is relevant for statistical surveys of extended regions in latent vector spaces which were filled by encoding or embedding Neural Networks. A particular reason for such a survey could be the study of the reaction of a Decoder to statistical vectors in an Autoencoder’s latent space. E.g. for creative purposes. During such surveys we want to fill extended regions of the latent space with statistical data points. More precisely: With points defined by vectors reaching out from the origin. The resulting point distribution does not need to be a homogeneous one, but it should cover the whole target volume somehow and should not miss certain sub-regions in it.

# Theoretically derived results for a uniform probability distribution per vector component

In my last post

Latent spaces – pitfalls of distributing points in multi dimensions – I – constant probability density per dimension

I derived some formulas for central properties of a very simple statistical vector distribution. We assumed that each component of the vectors could be created independently and with the help of a uniform, constant probability distribution: Each vector component was based on a random value taken from a defined real number interval [-b, b] with a constant and normalized probability density. Obviously, this process treats the components as statistically independent variables.

Resulting vector end points fill a quadratic area in a 2D-space or a cubic volume in 3D-space relatively well. See my last post for examples. The formulas revealed, however, that the end points of our vectors lie within a multi-dimensional spherical shell of an average radius <R>. This shell is relatively broad for small dimensions (N=2,3). But it gets narrower and narrower with a growing number of dimensions N ≥ 4.

In this post I will first test my formulas and approximations for a constant probability density in [-b, b] with the help of a numerical experiment. Afterward I discuss what kind of regions in a latent space we may miss even when we fill a sequence of growing cubes around the origin with statistical points based on our special vector distribution.

# Variational Autoencoder with Tensorflow – IX – taming Celeb A by resizing the images and using a generator

Another post in my series about options to handle the Kullback-Leibler [KL] loss of Variational Autoencoders [AEs] under the conditions of Tensorflows eager execution.

We still have to test the Python classes which we have so laboriously developed during the last posts. One of these classes, “VAE()”, supports a specific approach to control the KL-loss parameters during training and cost optimization by gradient descent: The class may use Tensorflow’s [TF 2] GradientTape-mechanism and the Keras function train_step() – instead of relying on Keras’ standard “add_loss()” functions.

Instead of recreating simple MNIST images of digits from ponts in a latent space I now want to train a VAE (with GradienTape-based loss control) to solve a more challenging task:

We want to create artificial images of naturally appearing human faces from randomly chosen points in the latent space of a VAE, which has been trained with images of real human faces.

Actually, we will train our VAE with images provided by the so called “Celeb A” dataset. This dataset contains around 200,000 images showing the heads of so called celebrities. Due to the number and size of its images this dataset forces me (due to my very limited hardware) to use a Keras Image Data Generator. A generator is a tool to transfer huge amounts of data in a continuous process and in form of small batches to the GPU during neural network training. The batches must be small enough such that the respective image data fit into the VRAM of the GPU. Our VAE classes have been designed to support a generator.

In this post I first explain why Celeb A poses a thorough test for a VAE. Afterwards I shall bring the Celeb A data into a form suitable for older graphics cards with small VRAM.

# Why do the Celeb A images pose a good test case for a VAE?

To answer the question we first have to ask ourselves why we need VAEs at all. Why do certain ML tasks require more than just a simple plain Autoencoder [AE]?

The answer to the latter question lies in the data distribution an AE creates in its latent space. An AE, which is trained for the precise reconstruction of presented images will use a sufficiently broad area/volume of the latent space to place different points corresponding to different imageswith a sufficiently large distance between them. The position in an AE’s latent space (together with the Encode’s and Decoder’s weights) encodes specific features of an image. A standard AE is not forced to generalize sufficiently during training for reconstruction tasks. On the contrary: A good reconstruction AE shall learn to encode as many details of input images as possible whilst filling the latent space.

However: The neural networks of a (V)AE correspond to a (non-linear) mapping functions between multi-dimensional vector spaces, namely

• between the feature space of the input data objects and the AE’s latent space
• and also between the latent space and the reconstruction space (normally with the same dimension as the original feature space for the input data).

This poses some risks whenever some tasks require to use arbitrary points in the latent space. Let us, e.g., look at the case of images of certain real objects in font of varying backgrounds:

During the AE’s training we map points of a high-dimensional feature-space for the pixel values of (colored) images to points in the multi-dimensional latent space. The target region in the latent space stemming from regions in the original feature-space which correspond to “reasonable” images displaying real objects may cover only a relatively thin, wiggled manifold within in the latent space (z-space). For points outside the curved boundaries of such regions in z-space the Decoder may not give you clear realistic and interpretable images.

The most important objectives of invoking the KL-loss as an additional optimization element by a VAE are

1. to confine the data point distribution, which the VAE’s Encoder part produces in the multidimensional latent space, around the origin O of the z-space – as far as possible symmetrically and within a very limited distance from O,
2. to normalize the data distribution around any z-point calculated during training. Whenever a real training object marks the center of a limited area in latent space then reconstructed data objects (e.g. images) within such an area should not be too different from the original training object.

I.e.: We force the VAE to generalize much more than a simple AE.

Both objectives are achieved via specific parameterized parts of the KL-loss. We optimize the KL-loss parameters – and thus the data distribution in the latent space – during training. After the training phase we want the VAE’s Decoder to behave well and smoothly for neighboring points in extended areas of the latent space:

The content of reconstructed objects (e.g. images) resulting from neighboring points within limited z-space areas (up to a certain distance from the origin) should vary only smoothly.

The KL loss provides the necessary smear-out effect for the data distribution in z-space.

During this series I have only shown you the effects of the KL-loss on MNIST data for a dimension of the latent space z_dim = 2. We saw the general confinement of z-points around the origin and also a confinement of points corresponding to different MNIST-numbers (= specific features of the original images) in limited areas. With some overlaps and transition regions for different numbers.

But note: The low dimension of the latent space in the MNIST case (between 2 and 16) simplifies the confinement task – close to the origin there are not many degrees of freedom and no big volume available for the VAE Encoder. Even a standard AE would be rather limited when trying to vastly distribute z-points resulting from MNIST images of different digits.

However, a more challenging task is posed by the data distribution, which a (V)AE creates e.g. of images showing human heads and faces with characteristic features in front of varying backgrounds. To get a reasonable image reconstruction we must assign a much higher number of dimensions to the latent space than in the MNIST case: z_dim = 256 or z_dim = 512 are reasonable values at the lower end!

Human faces or heads with different hair-dos are much more complex than digit figures. In addition the influence of details in the background of the faces must be handled – and for our objective be damped. As we have to deal with many more dimensions of the z-space than in the MNIST case a simple standard AE will run into trouble:

Without the confinement and local smear-out effect of the KL-loss only tiny and thin areas of the latent space will correspond to reconstructions of human-like “faces”. I have discussed this point in more detail also in the post
Autoencoders, latent space and the curse of high dimensionality – I

As a result a standard AE will NOT reconstruct human faces from randomly picked z-points in the latent space. So, an AE will fail on the challenge posed in the introduction of this post.

# Celeb A and the necessity to use a “generator” for the Celeb A dataset on graphics cards with small VRAM

I recommend to get the Celeb A data from some trustworthy Kaggle contributor – and not from the original Chinese site. You may find cropped images e.g. at here. Still check the image container and the images carefully for unwanted add-ons.

The Celeb A dataset contains around 200,000 images of the heads of celebrities with a resolution of 218×178 pixels. Each image shows a celebrity face in front of some partially complex background. The amount of data to be handled during VAE training is relatively big – even if you downscale the images. The whole set will not fit into the limited VRAM of older graphics cards as mine (GTX960 with 4 GB, only). This post will show you how to deal with this problem.

You may wonder why the Celeb A dataset poses a problem as the original data only consume about 1.3 GByte on a hard disk. But do not forget that we need to provide floating point tensors of size (height x width x 3 x 32Bit) instead of compressed integer based jpg-information to the VAE algorithm. You can do the math on your own. In addition: Working with multiple screens and KDE on Linux may already consume more than 1 GB of our limited VRAM.

## How can we deal with the Celeb A images on GPUs with limited VRAM ?

We use three tricks to work reasonably fast with the Celeb A data on a Linux systems with limited VRAM, but with around 32 GB or more standard RAM:

1. We first crop and downscale the images – in my case to 96×96 pixels.
2. We save a binary of a Numpy array of all images on a SSD and read it into the RAM during Jupyter experiments.
3. We then apply a so called Keras Image Data Generator to transfer the images to the graphics card when required.

The first point reduces the amount of MBytes per image. For basic experiments we do not need the full resolution.

The second point above is due to performance reasons: (1) Each time we want to work with a Jupyter notebook on the data we want to keep the time to load the data small. (2) We need the array data already in the system’s RAM to transfer them efficiently and in portions to the GPU.

A “generator” is a Keras tool which allows us to deliver input data for the VAE training in form of a continuously replenished dataflow from the CPU environment to the GPU. The amount of data provided with each transfer step to the GPU is reduced to a batch of images. Of course, we have to choose a reasonable size for such a batch. It should be compatible with the training batch size defined in the VAE-model’s fit() function.

A batch alone will fit into the VRAM whereas the whole dataset may not. The control of the data stream costs some overhead time – but this is better than not top be able to work at all. The second point helps to accelerate the transfer of data to the GPU significantly: A generator which sequentially picks data from a hard disk, transfers it to RAM and then to VRAM is too slow to get a convenient performance in the end.

Each time before we start VAE applications on the Jupyter side, we first fill the RAM with all image data in tensor-like form. From a SSD the totally required time should be small. The disadvantage of this approach is the amount of RAM we need. In my case close to 20 GB!

# Cropping and resizing Celeb A images

We first crop each of the original images to reduce background information and then resize the result to 96×96 px. D. Foster uses 128×128 px in his book on “Generative Deep Learning”. But for small VRAM 96×96 px is a bit more helpful.
I also wanted the images to have a quadratic shape because then one does not have to adjust the strides of
the VAE’s CNN Encoder and Decoder kernels differently for the two geometrical dimensions. 96 px in each dimension is also a good number as it allows for exactly 4 layers in the VAE’s CNNs. Each of the layers then reduces the resolution of the analyzed patterns by a factor of 2. At the innermost layer of the Encoder we deal with e.g. 256 maps with an extension of 6×6.

Cropping the original images is a bit risky as we may either cut some parts of the displayed heads/faces or the neck region. I decided to cut the upper part of the image. So I lost part of the hair-do in some cases, but this did not affect the ability to create realistic images of new heads or faces in the end. You may with good reason decide differently.

I set the edge points of the cropping region to

top=40, bottom = 0, left=0, right=178 .

This gave me quadratic pictures. But you may choose your own parameters, of course.

# A loop to crop and resize the Celeb A images

To prepare the pictures of the Celeb A dataset I used the PIL library.

```import os, sys, time
import numpy as np
import scipy
from glob import glob

import PIL as PIL
from PIL import Image
from PIL import ImageFilter

import matplotlib as mpl
from matplotlib import pyplot as plt
from matplotlib.colors import ListedColormap
import matplotlib.patches as mpat
```

A Juyter cell with a loop to deal with almost all CelebA images would then look like:

Jupyter cell 1

```dir_path_orig = 'YOUR_PATH_TO_THE_ORIGINAL_CELEB A_IMAGES'
dir_path_save = 'YOUR_PATH_TO_THE_RESIZED_IMAGES'

num_imgs = 200000 # the number of images we use

print("Started loop for images")
start_time = time.perf_counter()

# cropping corner positions and new img size
left  = 0;   top = 40
right = 178; bottom = 218
width_new  = 96
height_new = 96

# Cropping and resizing
for num in range(1, num_imgs):
jpg_name ='{:0>6}'.format(num)
jpg_orig_path = dir_path_orig + jpg_name +".jpg"
jpg_save_path = dir_path_save + jpg_name +".jpg"
im = Image.open(jpg_orig_path)
imc = im.crop((left, top, right, bottom))
#imc = imc.resize((width_new, height_new), resample=PIL.Image.BICUBIC)
imc = imc.resize((width_new, height_new), resample=PIL.Image.LANCZOS)
imc.save(jpg_save_path, quality=95)  # we save with high quality
im.close()

end_time = time.perf_counter()
cpu_time = end_time - start_time
print()
print("CPU-time: ", cpu_time)
```

Note that we save the images with high quality. Without the quality parameter PIL’s save function for a jpg target format would reduce the given quality unnecessarily and without having a positive impact on the RAM or VRAM consumption of the tensors we have to use in the end.

The whole process of cropping and resizing takes about 240 secs on my old PC without any parallelized operations on the CPU. The data were read from a standard old hard disk and not a SSD. As we have to make this investment of CPU time only once I did not care about optimization.

To prepare and save a huge Numpy array which contains all training images for our VAE we first need to define some parameters. I normally use 170,000 images for training purposes and around 10,000 for tests.

Jupyter cell 2

```# Some basic parameters
# ~~~~~~~~~~~~~~~~~~~~~~~~
INPUT_DIM          = (96, 96, 3)
BATCH_SIZE         = 128

# The number of available images
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
num_imgs = 200000  # Check with notebook CelebA

# The number of images to use during training and for tests
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
NUM_IMAGES_TRAIN  = 170000   # The number of images to use in a Trainings Run
#NUM_IMAGES_TO_USE  = 60000   # The number of images to use in a Trainings Run

NUM_IMAGES_TEST = 10000   # The number of images to use in a training Run

# for historic compatibility reasons of other code-fragments (the reader may not care too much about it)
N_ImagesToUse        = NUM_IMAGES_TRAIN
NUM_IMAGES           = NUM_IMAGES_TRAIN
NUM_IMAGES_TO_TRAIN  = NUM_IMAGES_TRAIN   # The number of images to use in a Trainings Run
NUM_IMAGES_TO_TEST   = NUM_IMAGES_TEST  # The number of images to use in a Test Run

# Define some shapes for Numpy arrays with all images for training
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
shape_ay_imgs = (N_ImagesToUse, ) + INPUT_DIM
print("Assumed shape for Numpy array with train imgs: ", shape_ay_imgs)

shape_ay_imgs_test = (NUM_IMAGES_TO_TEST, ) + INPUT_DIM
print("Assumed shape for Numpy array with test  imgs: ",shape_ay_imgs_test)
```

We also need to define some parameters to control the following aspects:

• Do we directly load Numpy arrays with train and test data?
• Do we load image data and convert them into Numpy arrays?
• From where do we load image data?

The following Jupyter cells help us:

Jupyter cell 3

```# Set parameters where to get the image data from
# ************************************************
# Use the cropped 96x96 HIGH-Quality images

# ~~~~~~~~~~~~~~~~~~~~~~~~~+
b_load_ay_from_saved = False     # True: Load prepared x_train and x_test Numpy arrays

# ~~~~~~~~~~~~~~~~~~~~~~

# Save newly calculated x_train, x_test-arrays in binary format onto disk
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
b_save_to_disk = False

# Paths
# ******

# Images on SSD
# ~~~~~~~~~~~~~
dir_path_load = 'YOUR_PATH_TO_HQ_DATA_ON_SSD/'    # high quality
else:
dir_path_load = 'YOUR_PATH_TO_HQ_DATA_ON_HD/'               #  low quality

# Images on slow HD
# ~~~~~~~~~~~~~~~~~~
# high quality on slow Raid
else:
# low quality on slow HD

# x_train, x_test arrays on SSD
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
dir_path_ay = 'YOUR_PATH_TO_Numpy_ARRAY_DATA_ON_SSD/'
path_file_ay_train = dir_path_ay + "celeba_200tsd_norm255_hq-x_train.npy"
path_file_ay_test  = dir_path_ay + "celeba_200tsd_norm255_hq-x_test.npy"
else:
path_file_ay_train = dir_path_ay + "celeba_200tsd_norm255_lq-x_train.npy"
path_file_ay_test  = dir_path_ay + "celeba_200tsd_norm255_lq-x_est.npy"

# x_train, x_test arrays on slow HD
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
dir_path_ay = 'YOUR_PATH_TO_Numpy_ARRAY_DATA_ON_HD/'
path_file_ay_train = dir_path_ay + "celeba_200tsd_norm255_hq-x_train.npy"
path_file_ay_test  = dir_path_ay + "celeba_200tsd_norm255_hq-x_test.npy"
else:
path_file_ay_train = dir_path_ay + "celeba_200tsd_norm255_lq-x_train.npy"
path_file_ay_test  = dir_path_ay + "celeba_200tsd_norm255_lq-x_est.npy"
```

You must of course define your own paths and names.
Note that the ending “.npy” defines the standard binary format for Numpy data.

# Preparation of Numpy array for CelebA images

In case that I want to prepare the Numpy arrays (and not load already prepared ones from a binary) I make use of the following straightforward function:

Jupyter cell 4

```def load_and_scale_celeba_imgs(start_idx, num_imgs, shape_ay, dir_path_load):

ay_imgs = np.ones(shape_ay, dtype='float32')
end_idx = start_idx + num_imgs

# We open the images and transform them into Numpy arrays
for j in range(start_idx, end_idx):
idx = j - start_idx
jpg_name ='{:0>6}'.format(j)
jpg_orig_path = dir_path_load + jpg_name +".jpg"
im = Image.open(jpg_orig_path)

# transfrom data into a Numpy array
img_array = np.array(im)
ay_imgs[idx] = img_array
im.close()

# scale the images
ay_imgs = ay_imgs / 255.

return ay_imgs
```

We call this function for training images as follows:

Jupyter cell 5

```# Load training images from SSD/HD and prepare Numpy float32-arrays
#               - (18.1 GByte of RAM required !! Int-arrays)
#               - takes around 30 to 35 Secs
# ************************************

# Prepare float32 Numpy array for the training images
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
start_idx_train = 1
print("Started loop for training images")
start_time = time.perf_counter()
num_imgs=NUM_IMAGES_TRAIN,
shape_ay=shape_ay_imgs_train,

end_time = time.perf_counter()
cpu_time = end_time - start_time
print()
print("CPU-time for array of training images: ", cpu_time)
print("Shape of x_train: ", x_train.shape)

# Plot an example image
plt.imshow(x_train[169999])
```

And for test images:

Jupyter cell 6

```# Load test images from SSD/HD and prepare Numpy float32-arrays
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

# Prepare Float32 Numpy array for test images
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
start_idx_test = NUM_IMAGES_TRAIN + 1

print("Started loop for test images")
start_time = time.perf_counter()
num_imgs=NUM_IMAGES_TEST,
shape_ay=shape_ay_imgs_test,

end_time = time.perf_counter()
cpu_time = end_time - start_time
print()
print("CPU-time for array of test images: ", cpu_time)
print("Shape of x_test: ", x_test.shape)

#Plot an example img
plt.imshow(x_test[27])
```

This takes about 35 secs in my case for the training images (170,000) and about 2 secs for the test images. Other people in the field use much lower numbers for the amount of training images.

If you want to save the Numpy arrays to disk:

Jupyter cell 7

```# Save the newly calculatd NUMPY arrays in binary format to disk
# ****************************************************************
print("Start saving arrays to disk ...")
np.save(path_file_ay_train, x_train)
print("Finished saving the train img array")
np.save(path_file_ay_test, x_test)
print("Finished saving the test img array")
```

If we wanted to load the Numpy arrays with training and test data from disk we would use the following code:

Jupyter cell 8

```# Load the Numpy arrays with scaled Celeb A directly from disk
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
print("Started loop for test images")
start_time = time.perf_counter()

end_time = time.perf_counter()
cpu_time = end_time - start_time
print()
print("Shape of x_train: ", x_train.shape)
print("Shape of x_test:  ", x_test.shape)
```

This takes about 2 secs on my system, which has enough and fast RAM. So loading a prepared Numpy array for the CelebA data is no problem.

# Defining the generator

Easy introductions to Keras’ ImageDataGenerators, their purpose and usage are given here and here.

ImageDataGenerators can not only be used to create a flow of limited batches of images to the GPU, but also for parallel operations on the images coming from some source. The latter ability is e.g. very welcome when we want to create additional augmented images data. The sources of images can be some directory of image files or a Python data structure. Depending on the source different ways of defining a generator object have to be chosen. The ImageDataGenerator-class and its methods can also be customized in very many details.

If we worked on a directory we might have to define our generator similar to the following code fragment

```    data_gen = ImageDataGenerator(rescale=1./255) # if the image data are not scaled already for float arrays
# class_mode = 'input' is used for Autoencoders
data_flow = data_gen.flow_from_directory(directory = YOUR_PATH_TO_ORIGINAL IMAGE DATA
#, target_size = INPUT_DIM[:2]
, batch_size = BATCH_SIZE
, shuffle = True
, class_mode = 'input'
, subset = "training"
)
```

This would allow us to read in data from a prepared sub-directory “YOUR_PATH_TO_ORIGINAL IMAGE DATA/train/” of the file-system and scale the pixel data at the same time to the interval [0.0, 1.0]. However, this approach is too slow for big amounts of data.

As we already have scaled image data available in RAM based Numpy arrays both the parameterization and the usage of the Generator during training is very simple. And the performance with RAM based data is much, much better!

So, how to our Jupyter cells for defining the generator look like?

Jupyter cell 9

```# Generator based on Numpy array for images in RAM
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
b_use_generator_ay = True
BATCH_SIZE    = 128
SOLUTION_TYPE = 3

if b_use_generator_ay:
# solution_type == 0 works with extra layers and add_loss to control the KL loss
# it requires the definition of "labels" - which are the original images
if SOLUTION_TYPE == 0:
data_gen = ImageDataGenerator()
data_flow = data_gen.flow(
x_train
, x_train
#, target_size = INPUT_DIM[:2]
, batch_size = BATCH_SIZE
, shuffle = True
#, class_mode = 'input'   # Not working with this type of generator
#, subset = "training"    # Not required
)
if ....
if ....

if SOLUTION_TYPE == 3:
data_gen = ImageDataGenerator()
data_flow = data_gen.flow(
x_train
#, x_train
#, target_size = INPUT_DIM[:2]
, batch_size = BATCH_SIZE
, shuffle = True
#, class_mode = 'input'   # Not working with this type of generator
#, subset = "training"    # Not required
)

```

Besides the method to use extra layers with layer.add_loss() (SOLUION_TYPE == 0) I have discussed other methods for the handling of the KL-loss in previous posts. I leave it to the reader to fill in the correct statements for these cases. In our present study we want to use a GradientTape()-based method, i.e. SOLUTION_TYPE = 3. In this case we do NOT need to pass a label-array to the Generator. Our gradient_step() function is intelligent enough to handle the loss calculation on its own! (See the previous posts).

So it is just

```        data_gen = ImageDataGenerator()
data_flow = data_gen.flow(
x_train
, batch_size = BATCH_SIZE
, shuffle = True
)
```

which does a perfect job for us.

In the end we will only need the following call when we want to train our VAE-model

```MyVae.train_myVAE(
data_flow
, b_use_generator = True
, epochs = n_epochs
, initial_epoch = INITIAL_EPOCH
)
```

to train our VAE-model. This class function in turn will internally call something like

```    self.model.fit(
data_flow   # coming as a batched dataflow from the outside generator
, shuffle = True
, epochs = epochs
, batch_size = batch_size # best identical to the batch_size of data_flow
, initial_epoch = initial_epoch
)

```

But the setup of a reasonable VAE-model for CelebA images and its training will be the topic of the next post.

# Conclusion

What have we achieved? Nothing yet regarding VAE results. However, we have prepared almost 200,000 CelebA images such that we can easily load them from disk into a Numpy float32 array with 2 seconds. Around 20 GB of conventional PC RAM is required. But this array can now easily be used as a source of VAE training.

Furthermore I have shown that the setup of a Keras “ImageDataGenerator” to provide the image data as a flow of batches fitting into the GPU’s VRAM is a piece of cake – at least for our VAE objectives. We are well prepared now to apply a VAE-algorithm to the CelebA data – even if we only have an old graphics card available with limited VRAM.

In the next post of this series

I show you the code for VAE-training with CelebA data. Afterward we will pick random points in the latent space and create artificial images of human faces.
Variational Autoencoder with Tensorflow – X – VAE application to CelebA images
People interested in data augmentation should have a closer look at the parameterization options of the ImageDataGenerator-class.

And last not least my standard statement as long as the war in Ukraine is going on:
Ceterum censeo: The worst fascist, war criminal and killer living today is the Putler. He must be isolated at all levels, be denazified and sooner than later be imprisoned. Long live a free and democratic Ukraine!

# Autoencoders, latent space and the curse of high dimensionality – I

Recently, I had to give a presentation about standard Autoencoders (AEs) and related use cases. Whilst preparing examples I stumbled across a well-known problem: The AE solved tasks as to reconstruct faces hidden in extreme noisy or leaky input images perfectly. But the reconstruction of human faces from arbitrarily chosen points in the so called “latent space” of a standard Autoencoder did not work well.

In this series of posts I want to discuss this problem a bit as it illustrates why we need Variational Autoencoders for a systematic creation of faces with varying features from points and clusters in the latent space. But the problem also raises some fundamental and interesting questions

• about a certain “blindness” of neural networks during training in general, and
• about the way we save or conserve the knowledge which a neural network has gained about patterns in input data during training.

This post requires experience with the architecture and principles of Autoencoders.

Note, 02/14/2023: I have revised and edited this post to get consistent with new insights from extended experiments with AEs and VAEs.

# Standard tasks for conventional Autoencoders

For preparing my talk I worked with relatively simple Autoencoders. I used Convolutional Neural Networks [CNNs] with just 4 convolutional layers to create the Encoder and Decoder parts of the Autoencoder. As typical applications I chose the following:

• Effective image compression and reconstruction by using a latent space of relatively low dimensionality. The trained AEs were able to compress input images into latent vectors with only few components and reconstruct the original image from the compressed format.
• Denoising of images where the original data were obscured by the superposition of statistical noise and/or statistically dropped pixels. (This is my favorite task for AEs which they solve astonishingly well.)
• Recolorization of images: The trained AE in this case transforms images with only gray pixels into colorful images.

Such challenges for AEs are discussed in standard ML literature. In a first approach I applied my Autoencoders to the usual MNIST and Fashion MNIST datasets. For the task of recolorization I used the Cifar 10 dataset. But a bit later I turned to the Celeb A dataset with images of celebrity faces. Just to make all of the tasks a bit more challenging.

# Standard Autoencoders and low dimensions of the latent space for (Fashion) MNIST and Cifar10 data

My Autoencoders excelled in all the tasks named above – both for MNIST, CELEB A and, regarding recolorization, CIFAR 10.

Regarding MNIST and MNIST/Fashion 4-layer CNNs for the Encoder and Decoder are almost an overkill. For MNIST the dimension z_dim of the latent space can be chosen to be pretty small:

z_dim = 12 gives a really good reconstruction quality of (test) images compressed to minimum information in the latent space. z_dim=4 still gave an acceptable quality and even with z_dim = 2 most of test images were reconstructed well enough. The same was true for the reconstruction of images superimposed with heavy statistical noise – such that the human eye could no longer guess the original information. For Fashion MNIST a dimension number 20 < z_dim < 40 gave good results. Also for recolorization the results were very plausible. I shall present the results in other blog posts in the future.

# Face reconstructions of (noisy) Celeb A images require a relative high dimension of the latent space

Then I turned to the Celeb A dataset. By the way: I got interested in Celeb A when reading the books of David Foster on “Generative Deep Learning” and of Tariq Rashi “Make Your First GANs with PyTorch” (see the complete references in the last section of this post).

The Celeb A data set contains images of around 200,000 faces with varying contours, hairdos and very different, in-homogeneous backgrounds. And the faces are displayed from very different viewing angles.

For a good performance of image reconstruction in all of the named use cases one needs to raise the number of dimensions of the latent space significantly. Instead of 12 dimensions of the latent space as for MNIST we now talk about 200 up to 1200 dimensions for CELEB A – depending on the task the AE gets trained for and, of course, on the quality expectations. For reconstruction of normal images and for the reconstruction of clear images from noisy input images higher numbers of dimensions z_dim ≥ 512 gave visibly better results.

Actually, the impressive quality for the reconstruction of test images of faces, which were almost totally obscured by the superimposition of statistical noise or the statistical removal of pixels after a self-supervised training on around 100,000 images surprised me. (Totalitarian states and security agencies certainly are happy about the superb face reconstruction capabilities of even simple AEs.) Part of the explanation, of course, is that 20% un-obscured or un-blurred pixels out of 30,000 pixels still means 6,000 clear pixels. Obviously enough for the AE to choose the right pattern superposition to compose a plausible clear image.

Note that we are not talking about overfitting here – the Autoencoder handled test images, i.e. images which it had never seen before, very well. AEs based on CNNs just seem to extract and use patterns characteristic for faces extremely effectively.

But how is the target space of the Encoder, i.e. the latent space, filled for Celeb A data? Do all points in the latent space give us images with well recognizable faces in the end?

# Face reconstruction after a training based on Celeb A images

To answer the last question I trained an AE with 100,000 images of Celeb A for the reconstruction task named above. The dimension of the latent space was chosen to be z_dim = 200 for the results presented below. (Actually, I used a VAE with a tiny amount of KL loss by a factor of 1.e-6 smaller than the standard Binary Cross-Entropy loss for reconstruction – to get at least a minimum confinement of the z-points in the latent space. But the results are basically similar to those of a pure AE.)

My somewhat reworked and centered Celeb A images had a dimension of 96×96 pixels. So the original feature space had a number of dimensions of 27,648 (almost 30000). The challenge was to reproduce the original images from latent data points created of test images presented to the Encoder. To be more precise:

After a certain number of training epochs we feed the Encoder (with fixed weights) with test images the AE has never seen before. Then we get the components of the vectors from the origin to the resulting points in the latent space (z-points). After feeding these data into the Decoder we expect the reproduction of images close to the test input images.

With a balanced training controlled by an Adam optimizer I already got a good resemblance after 10 epochs. The reproduction got better and very acceptable also with respect to tiny details after 25 epochs for my AE. Due to possible copyright and personal rights violations I do not dare to present the results for general Celeb A images in a public blog. But you can write me a mail if you are interested.

Most of the data points in the latent space were created in a region of 0 < |x_i| < 20 with x_i meaning one of the vector components of a z-point in the latent space. I will provide more data on the z-point distribution produced by the Encoder in later posts of this mini-series.

# Face reconstruction from randomly chosen points in the latent space

Then I selected arbitrary data points in the latent space with randomly chosen and uniformly distributed components 0 < |x_i| < boundary. The values for boundary were systematically enlarged.

Note that most of the resulting points will have a tendency to be located in outer regions of the multidimensional cube with an extension in each direction given by boundary. This is due to the big chance that one of the components will get a relatively high value.

Then I fed these arbitrary z-points into the Decoder. Below you see the results after 10 training epochs of the AE; I selected only 10 of 100 data points created for each value of boundary (the images all look more or less the same regarding the absence or blurring of clear face contours):

boundary = 0.5

boundary = 2.5

boundary = 5.0

boundary = 8.0

boundary = 10.0

boundary = 15.0

boundary = 20.0

boundary = 30.0

boundary = 50

This is more a collection of face hallucinations than of usable face images. (Interesting for artists, maybe? Seriously meant …).

So, most of the points in the latent space of an Autoencoder do NOT represent reasonable faces. Sometimes our random selection came close to a region in latent space where the result do resemble a face. See e.g. the central image for boundary=10.

From the images above it becomes clear that some arbitrary path inside the latent space will contain more points which do NOT give you a reasonable face reproduction than points that result in plausible face images – despite a successful training of the Autoencoder.

This result supports the impression that the latent space of well trained Autoencoders is almost unusable for creative purposes. It also raises the interesting question of what the distribution of “meaningful points” in the latent space really looks like. I do not know whether this has been investigated in depth at all. Some links to publications which prove a certain scientific interest in this question are given in the last section of this posts.

I also want to comment on an article published in the Quanta Magazine lately. See “Self-Taught AI Shows Similarities to How the Brain Works”. This article refers to “masked” Autoencoders and self-supervised learning. Reconstructing masked images, i.e. images with a superposition of a mask hiding/blurring pixels with a reasonably equipped Autoencoder indeed works very well. Regarding this point I totally agree. Also with the term “self-supervised learning”.

But to suggest that an Autoencoder with this (rather basic) capability reflects methods of the human brain is in my opinion a massive exaggeration. On the contrary, in my opinion an AE reflects a dumbness regarding the storage and usage of otherwise well extracted feature patterns. This is due to its construction and the nature of its mapping of image contents to the latent space. A child can, after some teaching, draw characteristic features of human faces – out of nothing on a plain white piece of paper. The Decoder part of a standard Autoencoder (in some contrast to a GAN) can not – at least not without help to pick a meaningful point in latent space. And this difference is a major one, in my opinion.

# A first interpretation – the curse of many dimensions of the latent space

I think the reason why arbitrary points in the multi-dimensional latent space cannot be mapped to images with recognizable faces is yet another effect of the so called “curse of high dimensionality”. But this time also related to the latent space.

A normal Autoencoder (i.e. one without the Kullback-Leibler loss) uses the latent space in its vast extension to produce points where typical properties (features) of faces and background are encoded in a most unique way for each of the input pictures. But the distinct volume filled by such points is a pretty small one – compared to the extensions of the high dimensional latent space. The volume of data points resulting from a mapping-transformation of arbitrary points in the original feature space to points of the latent space is of course much bigger than the volume of points which correspond to images showing typical human faces.

This is due to the fact that there are many more images with arbitrary pixel values already in the original feature space of the input images (with lets say 30000 dimensions for 100×100 color pixels) than images with reasonable values for faces in front of some background. The points in the feature space which correspond to reasonable images of faces (right colors and dominant pixel values for face features), is certainly small compared to the extension of the original feature space. Therefore: If you pick a random point in latent space – even within a confined (but multidimensional) volume around the origin – the chance that this point lies outside the particular volume of points which make sense regarding face reproduction is big. I guess that for z_dim > 200 the probability is pretty close to 1.

In addition: As the mapping algorithm of a neural Encoder network as e.g. CNNs is highly non-linear it is difficult to say how the boundary hyperplanes of mapping areas for faces look like. Complicated – but due to the enormous number of original images with arbitrary pixel values – we can safely guess that they enclose a rather small volume.

The manifold of data points in the z-space giving us recognizable faces in front of a reasonably separated background may follow a curved and wiggly “path” through the latent space. In principal there could even be isolated unconnected regions separated by areas of “chaotic reconstructions”.

I think this kind of argumentation line holds for standard Autoencoders and variational Autoencoders with a very small KL loss in comparison to the reconstruction loss (BCE (binary cross-entropy) or MSE).

# Why do Variational Autoencoders [VAEs] help?

The fist point is: VAEs reduce the total occupied volume of the latent space. Due to mu-related term in the Kullback-Leibler loss the whole distribution of z-points gets condensed into a limited volume around the origin of the latent space.

The second reason is that the distribution of meaningful points are smeared out by the logvar-term of the Kullback-Leibler loss.

Both effects enforce overlapping regions of meaningful standard Gaussian-like z-point distributions in the latent space. So VAEs significantly increase the probability to hit a meaningful z-point in latent space – if you chose points around the origin within a distance of “1” per coordinate (or vector component).

The total distance of a point and its vector in z-space has to be measured with some norm, e.g. the Euclidian one. Actually we should get meaningful reconstructions around a multidimensional sphere of radius “16”. Why this is reasonable will be discussed in forthcoming posts.

Please, also look at the series on the technical realization of VAEs in this blog. The last posts there prove the effects of the KL-loss experimentally for Celeb A data. Below you find a selection of images created from randomly chosen points in the latent space of a Variational Autoencoder with z_dim=200 after 10 epochs.

# Conclusion

Enough for today. Whilst standard Autoencoders solve certain tasks very well, they seem to produce very specific data distributions in the latent space for CelebA images: Only certain regions seem to be suitable for the reconstruction of “meaningful” images with human faces.

This problem may have its origin already in the feature space of the original images. Also there only a small minority of points represents humanly interpretable face images. This becomes obvious when you look at the vast amount of possible pixel values in a feature space of lets say 96x96x3 = 27,648. Each of these dimension can get a value between 0 and 255. This gives us more than 7 million combinations. Only a tiny fraction of these possible images will show reasonable faces in the center with a reasonably structured background around.

From a first experiment the chance of hitting a data point in latent space which gives you a meaningful image seems to be small. This result appears to be a variant of the curse of high dimensionality – this time including the latent space.

In a forthcoming post
Autoencoders, latent space and the curse of high dimensionality – II – a view on fragments and filaments of the latent space for CelebA images
we will investigate the z-point distribution in latent space with a variety of tools. And find that this distribution is fragmented and that the z-points for CelebA images are arranged in certain regions of the latent space. In addition we will get indications that the distribution contains filament-like structures.

https://towardsdatascience.com/ exploring-the-latent-space-of-your-convnet-classifier-b6eb862e9e55

Felix Leeb, Stefan Bauer, Michel Besserve,Bernhard Schölkopf, “Exploring the Latent Space of Autoencoders with
Interventional Assays”, 2022,
https://arxiv.org/abs/2106.16091v2 // https://arxiv.org/pdf/2106.16091.pdf
https://wiredspace.wits.ac.za/ handle/10539/33094?show=full
https://www.elucidate.ai/post/ exploring-deep-latent-spaces

Books:
T. Rashid, “GANs mit PyTorch selbst programmieren”, 2020, O’Reilly, dpunkt.verlag, Heidelberg, ISBN 978-3-96009-147-9
D. Foster, “Generatives Deep Learning”, 2019, O’Reilly, dpunkt.verlag, Heidelberg, ISBN 978-3-96009-128-8