Unsupervised learning networks

Department of Information Technology 1Soft Computing (ITC4256 )
Dr. C.V. Suresh Babu
Professor
Department of IT
Hindustan Institute of Science & Technology
Unsupervised learning networks

Action Plan
• Unsupervised Learning Networks
- Introduction to Kohonen Self-Organizing Feature Maps (KSOM)
- Rectangular grid computing
- Hexagonal grid computing
- KSOM architecture
- KSOM training algorithm
• Quiz at the end of session

Kohonen Self-Organizing Feature Maps (KSOM)
• Suppose if there are some pattern of arbitrary
dimensions, however, we need them in one
dimension or two dimensions.
• Then the process of feature mapping would be
very useful to convert the wide pattern space
into a typical feature space.
• There can be various topologies, however the
following two topologies are used the most:
- Rectangular Grid Topology
- Hexagonal Grid Topology

Rectangular Grid Topology
• This topology has 24 nodes in the distance-2 grid, 16 nodes in the distance-1 grid, and 8 nodes in the
distance-0 grid, which means the difference between each rectangular grid is 8 nodes.
• The winning unit is indicated by #.

Hexagonal Grid Topology
• This topology has 18 nodes in the distance-2 grid, 12 nodes in the distance-1 grid, and 6 nodes in the
distance-0 grid, which means the difference between each rectangular grid is 6 nodes.
• The winning unit is indicated by #.

KSOM - Architecture
• The architecture of KSOM is similar to that of the competitive
network.
• With the help of neighborhood schemes, discussed earlier, the
training can take place over the extended region of the network.

KSOM – Training Algorithm
Step 1 − Initialize the weights, the learning rate α and the neighborhood
topological scheme.
Step 2 − Continue step 3-9, when the stopping condition is not true.
Step 3 − Continue step 4-6 for every input vector x.
Step 4 − Calculate Square of Euclidean Distance for j = 1 to m
n m
D(j) = ∑ ∑ (xi − wij)2
i=1 j=1
Step 5 − Obtain the winning unit J where D j is minimum.

KSOM – Training Algorithm (Cont…)
Step 6 − Calculate the new weight of the winning unit by the following relation −
wij(new) = wij(old) + α[xi−wij(old)]
Step 7 − Update the learning rate α by the following relation −
α(t+1)=0.5αt
Step 8 − Reduce the radius of topological scheme.
Step 9 − Check for the stopping condition for the network.

Quiz - Questions
1. What are the 2 topologies used in KSOM?
2. The winning unit is indicated by ---------.
a) * b) $ c) # d) !
3. What parameters has to be initialized for the training algorithm?
4. What is done in step 8?
5. The architecture of KSOM is similar to that of the ------------ network.

Quiz - Answers
1. What are the 2 topologies used in KSOM?
i. Rectangular Grid Topology ii. Hexagonal Grid Topology
2. The winning unit is indicated by ---------.
a) * b) $ c) # d) !
3. What parameters has to be initialized for the training algorithm?
Weights, learning rate α and the neighbourhood topological scheme.
4. What is done in step 8?
Reduce the radius of topological scheme.
5. The architecture of KSOM is similar to that of the ------------ network.
Competitive

Action Plan
• Unsupervised Learning Networks (Cont…)
- Introduction to ART
- Operational principle of ART
- ART1 architecture
- ART1 training algorithm

Adaptive Resonance Theory (ART)
• This network was developed by Stephen Grossberg and Gail Carpenter in 1987.
• It is based on competition and uses unsupervised learning model.
• Basically, ART network is a vector classifier which accepts an input vector and classifies it into one of
the categories depending upon which of the stored pattern it resembles the most.

ART – Operational Principle
• The main operation of ART classification can be divided into the following
phases:
- Recognition phase
- Comparison phase
- Search phase

ART1 - Architecture
1. Computational Unit:
a. Input unit (F1 layer):
i. F1a layer Input portion
ii. F1b layer Interface
b. Cluster Unit (F2 layer)
c. Reset Mechanism
2. Supplement Unit:
• Two supplemental units namely, G1 and G2 is added along with reset unit, R.
• They are called gain control units.

ART1 – Architecture (Cont…)

ART1 – Architecture (Cont…)
Parameters Used:
• n − Number of components in the input vector
• m − Maximum number of clusters that can be formed
• bij − Weight from F1b to F2 layer, i.e. bottom-up weights
• tji − Weight from F2 to F1b layer, i.e. top-down weights
• ρ − Vigilance parameter
• ||x|| − Norm of vector x

ART1 – Training Algorithm
Step 1 − Initialize the learning rate, the vigilance parameter, and the weights as
follows −
α > 1 and 0 < ρ ≤ 1
0 < bij(0) < (α) / (α − 1 + n) and tij(0) = 1
Step 2 − Continue step 3-9, when the stopping condition is not true.
Step 3 − Continue step 4-6 for every training input.
Step 4 − Set activations of all F1a and F1 units as follows
F2 = 0 and F1a = input vectors
Step 5 − Input signal from F1a to F1b layer must be sent like
si = xi

ART1 – Training Algorithm (Cont…)
Step 6 − For every inhibited F2 node
yj = ∑ bijxi the condition is yj ≠ -1
i
Step 7 − Perform step 8-10, when the reset is true.
Step 8 − Find J for yJ ≥ yj for all nodes j.
Step 9 − Again calculate the activation on F1b as follows
xi = sitji
Step 10 − Now, after calculating the norm of vector x and vector s, we need to
check the reset condition as follows −
• If ||x||/ ||s|| < vigilance parameter ρ, then inhibit node J and go to step 7
• Else If ||x||/ ||s|| ≥ vigilance parameter ρ, then proceed further.

ART1 – Training Algorithm (Cont…)
Step 11 − Weight updating for node J can be done as follows −
bij(new) = (αxi) / (α − 1 + ||x||)
tij(new) = xi
Step 12 − The stopping condition for algorithm must be checked.

Quiz - Questions
1. ART network is a --------- classifier.
a) vector b) scalar c) linear d) non-linear
2. What are the 3 phases of the main operation of ART?
3. Name the 2 units of ART architecture.
4. What are the 3 components of computational unit?
5. What is the full form ART?

Quiz - Answers
1. ART network is a --------- classifier.
a) vector
2. What are the 3 phases of the main operation of ART?
i. recognition ii. comparison iii. search
3. Name the 2 units of ART architecture.
i. computational unit ii. Supplement unit
4. What are the 3 components of computational unit?
i. input unit ii. Cluster unit iii. reset mechanism
5. What is the full form ART?
Adaptive Resonance Theory

Action Plan
- Introduction to Radial Basis Function (RBF) network
- RBF architecture
- Hidden neural model
- Gaussian RBF
- RBF network parameters
- RBF learning algorithms

Radial Basis Function (RBF) Network
• A function is radial basis(RBF) if its output depends on (is a non-increasing
function of) the distance of the input from a given stored vector.
• The output of the red vector is “interpolated” using the three green vectors,
where each vector gives a contribution that depends on its weight and on its
distance from the red point.
• w1 < w3 < w2

RBF - Architecture
• One hidden layer with RBF activation functions.
• Output layer with linear activation function.

Hidden Neuron Model
• Hidden units use radial basis functions.
• The output depends on the distance of the input x from the center t.
• t is called center.
• is called spread.
• Center and spread are parameters.

Hidden Neurons
• A hidden neuron is more sensitive to data points near its center.
• For Gaussian RBF this sensitivity may be tuned by adjusting the spread ,
where a larger spread implies less sensitivity.

Gaussian RBF

Types of
• Multiquadrics:
• Inverse multiquadrics:
• Gaussian functions (most used):

RBF Network Parameters
• What do we have to learn for a RBF NN with a given architecture?
- The centers of the RBF activation functions.
- The spreads of the Gaussian RBF activation functions.
- The weights from the hidden to the output layer.
• Different learning algorithms may be used for learning the RBF network parameters.

RBF - Learning Algorithm 1
• Centers are selected at random.
• Spreads are chosen by normalization:
• Then the activation function of hidden neuron i becomes:
• Weights are computed by means of the pseudo-inverse method.

Learning Algorithm 1 - Summary
1. Choose the centers randomly from the training set.
2. Compute the spread for the RBF function using the normalization method.
3. Find the weights using the pseudo-inverse method.

Clustering algorithm for finding the centers :
• Initialization: tk(0) random k = 1, …, m1
• Sampling: draw x from input space .
• Similarity matching: find index of center closer to x.
• Updating: adjust centers.
• Continuation: increment n by 1, goto 2 and continue until no noticeable changes of centers occur.

Learning Algorithm 2 - Summary
Hybrid Learning Process:
• Clustering for finding the centers.
• Spreads chosen by normalization.
• LMS algorithm for finding the weights.

• Apply the gradient descent method for finding centers, spread and weights,
by minimizing the (instantaneous) squared error.
• Update for:
centers:
spread:
weights:

Quiz - Questions
1. ---------- units use radial basis functions.
2. A hidden neuron is more sensitive to data points near its ---------.
3. What are the types of ?
4. By what means weights are found in RBF learning algorithm 2 ?
5. How the centers are chosen in RBF learning algorithm 1 ?

Quiz - Answers
1. -------- units use radial basis functions.
Hidden
2. A hidden neuron is more sensitive to data points near its ---------.
center
3. What are the types of ?
i. multiquadrics ii. inverse multiquadrics iii. Gaussian functions
4. By what means weights are found in RBF learning algorithm 2 ?
LMS algorithm
5. How the centers are chosen in RBF learning algorithm 1 ?
Centers randomly chosen from training set.

Action Plan
- Introduction to Counter Propagation (CP) network
- CP architecture
- CP outstar and instar
- CP Operation

Counter Propagation Network
• CP algorithm consists of a input, hidden and
output layer.
• In this case the hidden layer is called the
Kohonen layer & the output layer is called the
Grossberg layer.
• The activation of this winner neuron is set to 1
& the activation of all other neurons in this
layer is set to 0.

Counter Propagation Network (Cont…)
Purpose:
• Fast and coarse approximation of vector mapping.
• Input vectors x are divided into clusters/classes.
• Each cluster of x has output y, which is (hopefully) the average of
for all x in that class.

Architecture: Simple Forward CPN

Network Architecture

Counter Propagation Network (Cont…)
1. Invented by Robert Hecht-Nielson, founder of HNC inc.
2. Consists of two opposing networks, one for learning a function, the other
for learning its inverse.
3. Each network has two layers:
- A Kohonen first layer that clusters inputs.
- An ‘outstar’ second layer to provide the output values for each cluster.

CP - Outstar and Instar
• An instar responds to a single input.
• An outstar produces a single (multi dimensional) output d when simulated with a binary value x.
• Biologically, outstar would be synaptic weights, while instar would have dendritic ones.
• It is common to refer to weights as ‘synaptic’.

CP - Outstar and Instar (Cont…)
• Variations can be possible by adding weights.

CP Operation
• An outstar neuron is associated with each cluster representative.
• Given an input, the winner is found.
• An outstar is then stimulated to give the output.
• Since these networks operate by recognizing input patterns in the first
layer, one would generally use lots of neurons in this layer.

Quiz - Questions
1. In CP network the hidden layer is called the --------- layer & the output
layer is called the --------- layer.
2. The activation of this winner neuron is set to 1 & the activation of all other
neurons in this layer is set to 0.
a) true b) false
3. ---------- vectors x are divided into clusters/classes.
a) input b) output c) outstar d) instar
4. CP network consist of two opposing networks, one for learning a ----------,
the other for learning its ----------.
5. Biologically, outstar would be -------- weights, while instar would have
---------- ones.

Quiz - Answers
1. Kohonen & Grossberg
2. a) true
3. a) input
4. function & inverse
5. synaptic & dendritic

Unsupervised learning networks

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