An Approach to Artificial Concept Learning Based on Human Concept Learning by Using Artificial Neural Networks - Advancing Artificial Intelligence Through Biological Process Applications - page 136

These two results explain why spurious pat-
terns are loaded into the network. Let us see it
with an example:

the load of a vector into the net is greater than
the corresponding in BH.
Since all associated vectors are vectors of N 2
components taking value in {-1, 1}, their norms
are equal, || Suppose that we want to get pat ter n

( 1,1,

= V for all v . This result implies
that what is actually stored in the network is the
orientation of the vectors associated to loaded
patterns.
From the above expression for the increment of
energy, and using that components of associated
vectors are either -1 or 1, the following expres-
sion for the decrease of energy when a pattern is
loaded is obtained:

A || E N

X = − − − loaded into a BH network.
Then, its associated matrix will be:

1, 1,1)

1 − 1 1 1 − 1

 

 

− 1 1 − 1 − 1 1

 

 

G = 1 − 1 1 1 − 1

X

 

1 − 1 1 1 − 1

 

 

− 1 1 − 1 − 1 1

 

and therefore the associated vector will be: (3.4)

1

−∆ E ( V ) = ( N − 2 d ( V , X ))

2

H

2

= (1, − 1,1,1, − − 1, 1,1, − − 1, 1,1,1,

X

− 1,1,1, − 1,1, − 1,1,1, − − 1, 1,1, − − 1, 1,1)

where

H d V is the Hamming distance between
vectors v and X .
After this explanation, we propose a solution
for this problem:

( , )

But, as can be easily verified, the vector
(1, − = − − has the same associated matrix

and vector than the original pattern X , that is,

1,1,1, 1)

The augmented pattern X , associated to X , is
defined by appending to X the possible values of
its components, that is, if

− = . By using the previous lemmas, we obtain
that the increment of energy of both X and -X is
the same, so these two vectors are those whose
energy decreases most:
1

A A

X X

 = { m ,  , m }

, then 1 L

ˆ

.

X = ( X , … , X , m , … , m }

1 N 1 L

∆ E X ( ) = − < A , A >=

X X

Particularly:

2

1

− < A , A >= ∆ − E ( X )

X − X

2

 = { 1,1} −

• In case of bipolar outputs, , and

ˆ

X = ( X , … , X , − 1,1)

This fact also implies that the corresponding
∆ W is the same for both vectors.
These results explain the well-known problem
of loading the opposite pattern of Hopfield's as-
sociative memory.
When using MREM, spurious patterns are
generated by the network in the same way. For ex-
ample, when we load the pattern X = (3,3,2,1,4,2),
also the pattern X 1 = (4,4,3,2,1,3) is loaded, but
also X 2 = (1,1,4,3,2,4), since all of them have the
same associated vector, and produce the same
decrease in the energy function. So, in MREM,
the number of spurious patterns appearing after

consequently it is . 1 N

• If , then

 = {1, … , L

ˆ

X = ( X , … , X ,1, 2, … . , L )

1 N

By making use of augmented patterns, the
problem of spurious patterns is solved, as stated
in the next result:

Lemma 3. The function Ψ that associates an
augmented pattern to its corresponding associated
vector is injective.

It can be shown that if augmented patterns are
used, the state V whose energy decreases most