Nvidia GPU-support of Tensorflow/Keras on Opensuse Leap 15

When you start working with Google's Tensorflow on multi-layer and "deep learning" artificial neural networks the performance of the required mathematical operations may sooner or later become important. One approach to better performance is the use of a GPU (or multiple GPUs) instead of a CPU. Personally, I am not yet in a situation where GPU support is really required. My experimental CNNs are too small, yet. But starting with Keras and Tensorflow is a good point to cover the use of a GPU on my Opensuse Leap 15 systems anyway. Actually, it is also helpful for some tasks in security related environments, too. One example is testing the quality of passphrases for encryption. With JtR you may gain a factor of 10 in performance. It is interesting, how much faster an old 960 GTX card will be for a simple Tensorflow test application than my i7 CPU.

I have used Nvidia GPUs almost all my Linux life. To get GPU support for Nvidia graphics cards you need to install CUDA in its present version. This is 10.1 in August 2019. You get download and install information for CUDA at
https://developer.nvidia.com/cuda-zone => https://developer.nvidia.com/cuda-downloads
For an RPM for the x86-64 architecture and Opensuse Leap see:

Installation of "CUDA" and "cudcnn"

You may install the downloaded RPM (in my "case cuda-repo-opensuse15-10-1-local-10.1.168-418.67-1.0-1.x86_64.rpm") via YaST. After this first step you in a second step install the meta-packet named "cuda", which is available in YaST at this point. Or just install all other packets with "cuda" in the name (with the exception of the source code and dev-packets) via YaST.

A directory "/usr/local/cuda" will be built; its entries are soft links to files in a directory "/usr/local/cuda-10.1".

Note the "include" and the "lib64" sub-directories! After the installation, also links should exist in the central "/usr/lib64"-directory pointing to the files in "/usr/local/cuda/lib64".

Note from the file-endings that the particular present version [Aug. 2019) of the files may be something like "10.1.168".

Another important point is that you need to install "cudnn" (cudnn-10.1-linux-x64-v7.6.2.24.tgz) - a Nvidia specific library for certain Deep Learning program elements, which shall be executed on Nvidia GPU chips. You get these files via "https://developer.nvidi.com/cudnn". Unfortunately, you must become member of the Nvidia developer community to get access to these special files. After you downloaded the tgz-file and expanded it, you find some directories "include" and "lib64" with relevant files. You just copy these files (as user root) into the directories "/usr/local/cuda/include" and "/usr/local/cuda/lib64", respectively. Check the owner/group and rights of the copied files afterwards and change them to root/root and standard rights - just as given for the other files in teh target directories.

The final step is the follwoing:
Create links by dragging the contents of "/usr/local/cuda/include" to "/usr/include" and chose the option "Link here". Do the same for the files of "/usr/local/cuda/lib64" with "/usr/lib64" as the target directory. If you look at the link-directories of the files now in "usr/include" and "usr/lib64" you see exactly which files were given by the CUDA and cudcnn installation.

Additional libraries
In case you want to use Keras it is recommended to install the "openblas" libraries including the development packages on the Linux OS level. On an Opensuse system just search for packages with "openblas" and install them all. The same is true for the h5py-libraries. In your virtual python environment execute:
< p style="margin-left:50px;"pip3 install --upgrade h5py

Problems with errors regarding missing CUDA libraries after installation

Two stupid things may happen after this straight-forward installation :

  • The link structure between "/usr/lib64" and the files in "/usr/local/cuda/include" and "/usr/local/cuda/lib64" may be incomplete.
  • Although there are links from files as "libcufftw.so.10" to something like "libcufftw.so.10.1.168" some libraries and TensorFlow components may expect additional links as "libcufftw.so.10.0" to "libcufftw.so.10.1.168"

Both points lead to error messages when I tried to use GPU related test statements on a PyDEV console or Jupyter cell. Watch out for error messages which tell you about errors when opening specific libraries! In the case of Jupyter you may find such messages on the console or terminal window from which you started your test.

A quick remedy is to use a file-manager as "dolphin" as user root, mark all files in "/usr/local/cuda/include" and "usr/local/cuda/lib64" and place them as (soft) links into "/usr/include" and "/usr/lib64", respectively. Then create additional links there for the required libraries "libXXX.so.10.0" to "libXXX.so.10.1.168", where "XXX" stands for some variable part of the file name.

A simple test with Keras and the mnist dataset

I assume that you have installed the packages for tensorflow, tensorflow-gpu (!) and keras with pip3 in your Python virtualenv. Note that the package "tensorflow-gpu" MUST be installed after "tensorflow" to make the use of the GPU possible.

Then a test with a simple CNN for the "mnist" datatset can deliver information on performance differences :

Cell 1 of a Jupyter notebook:

import time 
import tensorflow as tf
from keras import backend as K
from tensorflow.python.client import device_lib
from keras.datasets import mnist
from keras import models
from keras import layers
from keras.utils import to_categorical

# function to provide CPU/GPU information 
# ---------------------------------------
def get_CPU_GPU_details():
    print("GPU ? ", tf.test.is_gpu_available())

# information on available CPUs/GPUs
# --------------------------------------
if tf.test.is_gpu_available(
    print ("GPU is available")

# Setting a parameter GPU or CPU usage 
#gpu = False 
gpu = True
if gpu: 
    GPU = True;  CPU = False; num_GPU = 1; num_CPU = 1
    GPU = False; CPU = True;  num_CPU = 1; num_GPU = 0
num_cores = 6

# control of GPU or CPU usage in the TF environment
# -------------------------------------------------
# See the literature links at the article's end for more information  

config = tf.ConfigProto(intra_op_parallelism_threads=num_cores,
                        device_count = {'CPU' : num_CPU,
                                        'GPU' : num_GPU}, 

config.gpu_options.force_gpu_compatible = True
session = tf.Session(config=config)

# Loading the mnist datatset via Keras 
(train_images, train_labels), (test_images, test_labels) = mnist.load_data()
network = models.Sequential()
network.add(layers.Dense(512, activation='relu', input_shape=(28*28,)))
network.add(layers.Dense(10, activation='softmax'))
network.compile(optimizer='rmsprop', loss='categorical_crossentropy', metrics=['accuracy'])
train_images = train_images.reshape((60000, 28*28))
train_images = train_images.astype('float32') / 255
test_images = test_images.reshape((10000, 28*28))
test_images = test_images.astype('float32') / 255
train_labels = to_categorical(train_labels)
test_labels = to_categorical(test_labels)

Output of the code in cell 1:

GPU is available
GPU ?  True
[name: "/device:CPU:0"
device_type: "CPU"
memory_limit: 268435456
locality {
incarnation: 17801622756881051727
, name: "/device:XLA_GPU:0"
device_type: "XLA_GPU"
memory_limit: 17179869184
locality {
incarnation: 6360207884770493054
physical_device_desc: "device: XLA_GPU device"
, name: "/device:XLA_CPU:0"
device_type: "XLA_CPU"
memory_limit: 17179869184
locality {
incarnation: 7849438889532114617
physical_device_desc: "device: XLA_CPU device"
, name: "/device:GPU:0"
device_type: "GPU"
memory_limit: 2115403776
locality {
  bus_id: 1
  links {
incarnation: 4388589797576737689
physical_device_desc: "device: 0, name: GeForce GTX 960, pci bus id: 0000:01:00.0, compute capability: 5.2"

Note the control settings for GPU usage via the parameter gpu and the variable "config". If you do NOT want to use the GPU execute

config = tf.ConfigProto(device_count = {'GPU': 0, 'CPU' : 1})

Information on other control parameters which can be used together with "tf.ConfigProto" is provided here:

Cell 2 of a Jupyter notebook for performance measurement during training:

start_c = time.perf_counter()
with tf.device("/GPU:0"):
    network.fit(train_images, train_labels, epochs=5, batch_size=30000)
end_c = time.perf_counter()
if CPU: 
    print('Time_CPU: ', end_c - start_c)  
    print('Time_GPU: ', end_c - start_c)  

Output of the code in cell 2 :

Epoch 1/5
60000/60000 [==============================] - 0s 3us/step - loss: 0.5817 - acc: 0.8450
Epoch 2/5
60000/60000 [==============================] - 0s 3us/step - loss: 0.5213 - acc: 0.8646
Epoch 3/5
60000/60000 [==============================] - 0s 3us/step - loss: 0.4676 - acc: 0.8832
Epoch 4/5
60000/60000 [==============================] - 0s 3us/step - loss: 0.4467 - acc: 0.8837
Epoch 5/5
60000/60000 [==============================] - 0s 3us/step - loss: 0.4488 - acc: 0.8726
Time_GPU:  0.7899935730001744

Now change the following lines in cell 1

gpu = False 
#gpu = True 

Executing the code in cell 1 and cell 2 then gives:

Epoch 1/5
60000/60000 [==============================] - 0s 6us/step - loss: 0.4323 - acc: 0.8802
Epoch 2/5
60000/60000 [==============================] - 0s 7us/step - loss: 0.3932 - acc: 0.8972
Epoch 3/5
60000/60000 [==============================] - 0s 6us/step - loss: 0.3794 - acc: 0.8996
Epoch 4/5
60000/60000 [==============================] - 0s 6us/step - loss: 0.3837 - acc: 0.8941
Epoch 5/5
60000/60000 [==============================] - 0s 6us/step - loss: 0.3830 - acc: 0.8908
Time_CPU:  1.9326397939985327

Thus the GPU is faster by a factor of 2.375 !
At least for the chosen batch size of 30000! You should play a bit around with the batch size to understand its impact.
2.375 is not a big factor - but I have a relatively old GPU (GTX 960) and a relatively fast CPU i7-6700K mit 4GHz Taktung: So I take what I get 🙂 . A GTX 1080Ti would give you an additional factor of around 4.

Watching GPU usage during Python code execution

A CLI command which gives you updated information on GPU usage and memory consumption on the GPU is

nvidia-smi -lms 250

It gives you something like

Mon Aug 19 22:13:18 2019       
| NVIDIA-SMI 418.67       Driver Version: 418.67       CUDA Version: 10.1     |
| GPU  Name        Persistence-M| Bus-Id        Disp.A | Volatile Uncorr. ECC |
| Fan  Temp  Perf  Pwr:Usage/Cap|         Memory-Usage | GPU-Util  Compute M. |
|   0  GeForce GTX 960     On   | 00000000:01:00.0  On |                  N/A |
| 20%   44C    P0    33W / 160W |   3163MiB /  4034MiB |      1%      Default |
| Processes:                                                       GPU Memory |
|  GPU       PID   Type   Process name                             Usage      |
|    0      4124      G   /usr/bin/X                                   610MiB |
|    0      4939      G   kwin_x11                                      54MiB |
|    0      4957      G   /usr/bin/krunner                               1MiB |
|    0      4959      G   /usr/bin/plasmashell                         195MiB |
|    0      5326      G   /usr/bin/akonadi_archivemail_agent             2MiB |
|    0      5332      G   /usr/bin/akonadi_imap_resource                 2MiB |
|    0      5338      G   /usr/bin/akonadi_imap_resource                 2MiB |
|    0      5359      G   /usr/bin/akonadi_mailfilter_agent              2MiB |
|    0      5363      G   /usr/bin/akonadi_sendlater_agent               2MiB |
|    0      5952      C   /usr/lib64/libreoffice/program/soffice.bin    38MiB |
|    0      8240      G   /usr/lib64/firefox/firefox                     1MiB |
|    0     13012      C   /projekte/GIT/ai/ml1/bin/python3            2176MiB |
|    0     14233      G   ...uest-channel-token=14555524607822397280    62MiB |

During code execution some of the displayed numbers - e.g for GPU-Util, GPU memory Usage - will start to vary.


http://www.science.smith.edu/dftwiki/index.php/Setting up Tensorflow 1.X on Ubuntu 16.04 w/ GPU support


Der nächste Schritt ….

Die Grenzen zur Beeinträchtigung der digitalen Privatsphäre werden (erwartungsgemäß) weiter ausgelotet:
Unsere Sicherheitsorgane nicht mehr nur als Hacker ?.?.? -- weil Hacking wohl zu mühsam ist und (erwartungsgemäß) nicht überall zum Erfolg führt?

Als nunmehr 60-Jähriger und Werte-Konservativer wundere ich mich über gar nichts mehr. Ich habe an der Schule noch gelernt, wie essentiell eine Beschneidung der Möglichkeiten von Inlandsgeheimdiensten aufgrund der Erfahrungen aus dem sog. Dritten Reich ist - und dass es nach unserer Verfassung zu keiner Verquickung geheimdienstlicher und polizeilicher Tätigkeiten kommen solle und dürfe. Und dass ein richterlicher Beschluss unbedingte Grundlage jeder Verletzung der Privatsphäre sein muss. Nun erleben wir, wie im Namen der Sicherheit von den Bewahrern der Sicherheit "weiter" gedacht wird. Dass sich große internationale IT-Monopolisten um Privatsphäre und die dt. Verfassung bislang schon wenig scherten, überraschte wenig; von führenden Politikern und Ministerien muss man als Demokrat dagegen mehr erwarten dürfen.

Früher verhinderte die Umsetzung solcher Ideen, wie sie in den in den Artikeln behandelten Gesetzentwürfen ausgeführt werden, allein schon der Aufstand der Rechtschaffenen in der FDP und der SPD. Wo bleibt deren Aufschrei heute? Die SPD fragt sich wahrscheinlich in Arbeitskreisen auf lokaler Ebene erstmal, ob sie in einer zunehmend digitalisierten Welt überhaupt leben wolle - kein Witz, das wurde mir von einem Ortsvorstand genau so in einem Biergarten gesagt - ... und wenn, ob man dafür dann nicht ein Tripel statt eines Duos für die Parteiführung benötige ... Hr. Lindner von der FDP würde wohl antworten, dass man solche komplexen Themen nur und ausschließlich den Profis überlassen muss - und nicht der Parteipolitik ...

Denkt man IT-Sicherheit und IT-Privatsphäre durch, so wird schnell klar, welche extreme Bedeutung der Unversehrtheit privat oder professionell genutzter IT-Systeme zukommt. Ist die nicht mehr gegeben, so gibt es schlicht und einfach gar keine Privatsphäre mehr - auch nicht bei Einsatz von auf Sicherheit getrimmten Linux-Systemen und voll-verschlüsselter System-/Daten-Platten bzw. -Partitionen oder gar der temporären Entfernung von Datenträgern aus den Systemen bei längerer Abwesenheit. Selbst wenn man als vorsichtiger Zeitgenosse (z.B. als Investigativ-Journalist) seine genutzten Linux-Betriebssysteme und seine Daten nach einer solchen längeren Abwesenheit nur aus voll-verschlüsselten Backups mit prüfbaren Hash-Summen rekonstruieren würde, die man zuvor an sicheren Orten platziert hätte, würde einen das gegenüber unbefugten Eindringlingen in Wohnung oder Home-Office nicht wirklich schützen. Da die Eindringlinge - egal welcher Coleur - ja womöglich Zugriff auf die Hardware der genutzten Systeme (wie PCs, Server) bekämen, könnten sie auch diese manipulieren. Das ist u.U. sogar einfacher und für den Betroffenen schwerer zu entdecken als Manipulationen an Software. Hatten uns die Amerikaner nicht vor kurzer Zeit genau davor im Zshg. mit Huwai und anderen chinesischen Herstellern gewarnt? Nun, das Volk der Dichter und Denker erweist sich als lernfähig. Tja, es scheint, als mangelte es selbst den Herren Huxley und Orwell an hinreichender Phantasie für die Wirklichkeit ...

The moons dataset and decision surface graphics in a Jupyter environment – VI – Kernel-based SVC algorithms

We continue with our article series on the moons dataset as an entry point into "Machine Learning":

The moons dataset and decision surface graphics in a Jupyter environment – I
The moons dataset and decision surface graphics in a Jupyter environment – II – contourplots
The moons dataset and decision surface graphics in a Jupyter environment – III – Scatter-plots and LinearSVC
The moons dataset and decision surface graphics in a Jupyter environment – IV – plotting the decision surface
The moons dataset and decision surface graphics in a Jupyter environment – V – a class for plots and some experiments

The moons dataset and the related classification problem posed interesting challenges for us as beginners in ML:

Most people starting with ML have probably studied the topic of linear classification to separate distinct data sets by a linear decision surface (hyperplane) on data (y, X) with X=(x1,x2) defining points in a 2-dim space.

The SVM-approach studied in this article series follows a so called "soft-margin" classification: It tries to maximize the distances of the decision surface to the data points whilst it at the same time tries to reduce the number of points which violate the separation, i.e. points which end up on the wrong side of the decision hyperplane. This approach is controlled by so called hyper-parameters as a parameter "C" in the LinearSVC algorithm. If we just used LinearSVC on the original (x1,x2) data plane the hyperplane would be some linear line.

Unfortunately, for the moons dataset we must to overcome the fundamental problem that a linear classification approach to define a decision surface between the two data clusters is insufficient. (We have confirmed this in our last article by showing that even a quadratic approach does not give any reasonable decision surface.) We circumvented this problem by a trick - namely a polynomial extension of the parameter space (x1,x2) via a SciKit-Learn function "PolynomialFeatures()".

We also had to tackle the technical problem of writing a simple Python class for creating plots of a decision surface in a 2 dim parameter space (x1,x2). After having solved this problem in the last articles it became easy to apply different or differently parameterized algorithms to the data set and display the results graphically. The functionalities of the SciKit and numpy libraries liberated us from dealing with complicated mathematical tasks. We also saw that using the "Pipeline" function helped to organize the sequential operations on the data sets - here transforming data, scaling data and training of a chosen algorithm on the data - in a very comfortable way.

In this article we want to visualize results of some other SVM approaches, which make use of the called "Kernel trick". We shall explain this briefly and then use functions provided by SciKit to perform further experiments. On the Python side we shall learn, how to measure required computational times of the different algorithms.

Artificial polynomial feature extension

So far we moved around the hurdle of non-linearity in combination with LinearSVC by a costly trick: We have extended the original 2-dimensional parameter space X(x1, x2) of our data points [y, X] artificially by more X-dimensions. We proclaimed that the distribution of data points is actually given in a multidimensional space constructed by a polynomial transformation: Each new and additional dimension is given by a term f*(x1**n)*(x2**m) with n+m = D, the degree of a polynomial function of x1 and x2.

This works as if the results "y" depended on further properties expressed by polynomial combinations of x1 and x2. Note that a 2-dim space (x1,x2) thus may be transformed into a 5-dimensional space with axes for x1, x2, x1**2, a*x1*x2, x**2. A data point (x1,x2) is transformed to a vector P(x1,x2) = [p_1=x1, p_2=x2, p_3=x1**2, p_4=a*x1*x2, p_5=x2**2]. Actually, for a broad class of problems it is enough to look at the 3-dim transformation space P([x1,x2])=[p_1=x1**2, p_2=f*x1*x2, p_3=x2**2].

In such a higher dimensional space we might actually find a "linear" hyperplane which allows for a suitable separation for the data clusters belonging to 2 different values of y=y(x1,x2) - here y=0 and y=1. The optimization algorithm then determines a suitable parameter vector (Theta= [theta_0, theta_1, ... theta=n]), describing an optimal linear surface with respect to the distance of the data points to this surface. If we omit details then the separation surface is basically described by some scalar product of the form

theta_0*1 + theta1*p_1 + theta2*p_2 + ... + theta_D * p_D = const.

Our algorithm calculates and optimizes the required theta-values.

Note that the projection of this hyperplane into the original (x1,x2)-feature-space becomes a non-linear hyperplane there. See the book of S. Raschka "Python machine Learning", 20115, PACKT Publishing, chapter 3 for a nice example).

I cannot go into mathematical details in this article series. Nevertheless, this is a raw description of what we have done so far. But note, that there are other methods to extend the parameter space of the original data points to higher dimensions. The extension by the use of polynomials is just one of many approaches.

Why is a dimensional extension by polynomials computationally costly?

The soft margin optimization is a so called quadratic problem with linear constraints. To solve it you have both to transform all points into the higher dimensional space, i.e. to determine point coordinates there, and then determine distances in this space to a hyperplane with varying parameters.

This means we have at least to perform 2*D different calculations of the powers of the individual original coordinates of our input data points. As the power operation itself requires CPU-time depending on D the coordinate transformation operations vary with the

CPU-time ∝ number of points x the dimension of the original space x D**2

The "Kernel Trick"

In a certain formulation of the optimization problem the equation which determines the optimal linear separation hyperplane is governed by scalar products of the transformed vectors T(P(a1,a2)) * P(b1,b2) for all given data points a(x1=a1, x2=a2) and b(x1=b1,x2=b2), with T representing the transpose operation.

Now, instead of calculating the scalar product of the transformed vectors we would like to use a simpler "kernel" function

K(a, b = T(P(a)) * P(b)

It can indeed be shown that such a function K, which only operates on the lower dimensional space (!), really exists under fairly general conditions.

Kernel functions, which are typically used in classification problems are:

  • Polynomial kernel : K(a, b) = [ f*P(a) * b + g]**D, with D = polynomial degree of a polynomial transformation
  • Gaussian RBF kernel: K(a, b) = exp(- g * || a - b ||**2 ), which also corresponds to an extension into a transformed parameter space of very high dimension (see below).

You can do the maths for the number and complexity operations for the polynomial kernel on your own. It is easy to see that it costs much less to perform a scalar product in the lower dimensional space and calculate the n-the power of the result just once - instead of transforming two points by around 2*D different power operations on the individual coordinates and then performing a scalar product in the higher dimensional space:

The difference in CPU costs between a non-kernel based polynomial extension and a kernel based grows quadratically, i.e. with D**2.

All good ?
Although it seems that the kernel trick saves us a lot of CPU-time, we also have to take into account convergence of the optimization process in the higher dimensional space. All in all the kernel trick works best on small complex datasets - but it may get slow on huge datasets. See for a discussion in the book "Hands on-On Machine Learning with Scikit-Learn and TensorFlow" of A.Geron,(2017, O'Reilly), chapter 5.

Gaussian RBF kernel

The Gaussian RBF kernel transforms the original feature space (x1,x2) into an extended multidimensional by a different approach: It looks at the similarity of a target point with selected other points - so called "landmarks":

A new feature (=dimension) value is calculated by the Gaussian weight of the distance of a data point to one of the selected landmarks.

The number of landmarks can be chosen to be equal to the number N of all (other) data points in the training set. Thus we would add N-1 new dimensions - which would be a large number. The transformations operations for the coordinates of the data points in the original space (x1,x2) would, therefore, be numerous, too.

However, the Gaussian kernel enhances computational efficiency by large factors: it works on the lower dimensional parameter space, only, and determines the distance of data point pairs there! And still gives the correct results in the higher dimensional space for a linear separation interface there.

It is clear that the width of the Gaussian function is an important ingredient in this approach; this is controlled by the hyper-parameter "g" of the algorithm.

How to measure computational time?

This is relatively simple. We need to import the module "time". It includes a suitable function "perf_counter()". See: docs.python.org - perf_counter

We have to call it before a statement whose duration we want to measure and afterwards. The difference gives the CPU-time needed in fractions of a second. See below for the application.

Quadratic dependency of CPU time on the polynomial degree without the kernel trick

Let us measure a time series fro our standard polynomial approach. In our moons-notebook from the last session we add a cell and execute the following code:

And the plot looks like :

We recognize the expected quadratic behavior.

Polynomial kernel - almost constant low CPU-time independent of the polynomial degree

Let us now compare the output of the approach PolynomialsFeature + LinearSVC to an approach with the polynomial kernel. SciKit-learn provides us with an interface "SVC" to the kernel based algorithms. We have to specify the type of kernel besides other parameters. We execute the code in the following 2 cells to get a comparison for a polynomial of degree 3:

You saw that we specified a kernel-type "poly" in the second cell ?
The plots - first for LinearSVC and then for the polynomial kernel look like

We see a difference in the shape of separation line. And we notice already a slightly better performance for the kernel based algorithm.

Now let us prepare a similar time series for the kernel based approach:

The time series looks wiggled - but note that all numbers are below 1.3 msec ! What a huge difference!

Plot for the Gaussian RBF Kernel

Just to check how the separation surface looks like for the Gaussian kernel, we do the following experiment; note that we specify a kernel named "rbf":

Oops, a very different surface in comparison to what we have seen before.
But: Even a minimum change of gamma gives us yet another surface:

Funny, isn't it? We learn again that algorithms are sensitive!

Enough for today!


We have learned a bit about the so called kernel trick in the SVM business. Again, SciKit-Learn makes it very simple for us to make use of kernel based algorithms via an SVC-interface: different kernels and related feature space extension methods can be defined as a parameter.

We saw that a "poly"-kernel based approach in comparison to LinearSVC saves a lot of CPU-time when we use high order polynomials to extend the feature space to related higher dimensions.

The Gaussian RBF-kernel which extends the feature space by adding dimension based on weighted distances between data points proved to be interesting: It constructs a very different separation surface in comparison to polynomial approaches. We saw that RBF-kernel reacts sensitively to its configuration parameter "gamma" - i.e the width of the Gaussian weighing the similarity influence of other points.

Again we saw that in regions of the (x1,x2)-plane where no test data were provided the algorithms may predict very different memberships of new data points to either of the two moon clusters. Such extrapolations may depend on (small) parameter changes for the algorithms.