September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
1
Visual Illusions demonstrate how we perceive an “interpreted version” of
the incoming light pattern rather that the exact pattern itself.
Visual Illusions
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
2
Visual Illusions
He we see that the squares A and B from the previous image actually have
the same luminance (but in their visual context are interpreted differently).
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
3
How do NNs and ANNs work?
•
NNs are able to
learn
by
adapting their
connectivity patterns
so that the organism
improves its behavior in terms of reaching certain
(evolutionary) goals.
•
The strength of a connection, or whether it is
excitatory or inhibitory, depends on the state of a
receiving neuron’s
synapses
.
•
The NN achieves
learning
by appropriately
adapting the states of its synapses.
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
4
An Artificial Neuron
x
1
x
2
x
n
…
W
i,1
W
i,2
…
W
i,n
x
i
neuron i
net input signal
synapses
output
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
5
The Activation Function
One possible choice is a
threshold function
:
The graph of this function looks like this:
1
0
f
i
(net
i
(t))
net
i
(t)
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
6
Binary Analogy: Threshold Logic Units (TLUs)
x
1
x
2
x
3
w
1
=
w
2
=
w
3
=
=
Example:
x
1
x
2
x
3
1
1

1
1.5
TLUs in technical systems are similar to the
threshold neuron model, except that TLUs only
accept binary inputs (0 or 1).
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
7
Binary Analogy: Threshold Logic Units (TLUs)
x
1
x
2
w
1
=
w
2
=
=
Yet another example:
x
1
x
2
XOR
Impossible! TLUs can only realize
linearly separable
functions.
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
8
Linear Separability
A function f:{0, 1}
n
{0, 1} is linearly separable if the
space of input vectors yielding 1 can be separated
from those yielding 0 by a
linear surface
(
hyperplane
) in n dimensions.
Examples (two dimensions):
1
0
1
1
x
2
x
1
0
1
0
1
1
0
0
1
x
2
x
1
0
1
0
1
linearly separable
linearly inseparable
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
9
Linear Separability
To explain linear separability, let us consider the
function f:
R
n
{0, 1} with
where x
1
, x
2
, …, x
n
represent real numbers.
This is the exactly the function that our threshold
neurons use to compute their output from their inputs.
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
10
Linear Separability
Input space in the two

dimensional case (n = 2):
w
1
= 1, w
2
= 2,
= 2
x
1
1
2
3

3

2

1
x
2
1
2
3

3

2

1
0
1
w
1
=

2, w
2
= 1,
= 2
x
1
1
2
3

3

2

1
x
2
1
2
3

3

2

1
0
1
w
1
=

2, w
2
= 1,
= 1
x
1
1
2
3

3

2

1
x
2
1
2
3

3

2

1
0
1
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
11
Linear Separability
So by varying the weights and the threshold, we can
realize
any linear separation
of the input space into
a region that yields output 1, and another region that
yields output 0.
As we have seen, a
two

dimensional
input space
can be divided by any straight line.
A
three

dimensional
input space can be divided by
any two

dimensional plane.
In general, an
n

dimensional
input space can be
divided by an (n

1)

dimensional plane or hyperplane.
Of course, for n > 3 this is hard to visualize.
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
12
Linear Separability
Of course, the same applies to our original function f
of the TLU using binary input values.
The only difference is the restriction in the input
values.
Obviously, we cannot find a straight line to realize the
XOR function:
1
0
0
1
x
2
x
1
0
1
0
1
In order to realize XOR with TLUs, we need to
combine multiple TLUs into a network.
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
13
Multi

Layered XOR Network
x
1
x
2
x
1
x
2
x
1
x
2
1

1
0.5

1
1
0.5
1
1
0.5
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
14
Capabilities of Threshold Neurons
What can threshold neurons do for us?
To keep things simple, let us consider such a neuron
with two inputs:
The computation of this neuron can be described as
the inner product of the
two

dimensional vectors
x
and
w
i
, followed by a threshold operation.
x
1
x
2
W
i,1
W
i,2
x
i
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
15
The Net Input Signal
The
net input signal
is the sum of all inputs after
passing the synapses:
This can be viewed as computing the
inner product
of the vectors
w
i
and
x
:
where
is the
angle
between the two vectors.
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
16
Capabilities of Threshold Neurons
Let us assume that the threshold
= 0 and illustrate the
function computed by the neuron for sample vectors
w
i
and
x
:
Since the inner product is positive for

90
90
, in this
example the neuron’s output is 1 for any input vector
x
to the
right of or on the dotted line, and 0 for any other input vector.
w
i
first vector component
second vector component
x
September 14, 2010
Neural Networks
Lecture 3: Models of Neurons and Neural Networks
17
Capabilities of Threshold Neurons
By choosing appropriate weights
w
i
and threshold
we can place the
line
dividing the input space into
regions of output 0 and output 1in
any position and
orientation
.
Therefore, our threshold neuron can realize any
linearly separable
function
R
n
{0, 1}.
Although we only looked at two

dimensional input,
our findings apply to
any dimensionality n
.
For example, for n = 3, our neuron can realize any
function that divides the three

dimensional input
space along a two

dimension plane.
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