MACHINE LEARNING TECHNIQUES - LASA

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the resulting distribution is a valid probability, i.e.
K
∑
i=
1
P = 1
resolved through Lagrange and gives the following update step:
i
. Such an optimization problem can be
( + 1 ) = ( ) − ( ) ( | ) − ( )
Pi n Pi n α n ⎡⎡⎣⎣P i x Pi
n ⎤⎤⎦⎦
(6.63)
where n is the update step and α is the learning rate.
To speed up the computation, the updating of the above parameters is then limited to a small
neighborhood around the “winning node”, i.e. the pdf i which has the largest posterior
probability ( | )
pi x . The distribution for a given point x is then approximated by a mixture of a
small number of nodes at one time, i.e.
with η a neighborhood around the winner.
( ) ( θ )
pˆ x ≈ ∑ p x|
P
(6.64)
i∈η
i i i
Learning thus proceeds as in Kohonen. At each iteration step n, one picks a data point at
random, selects the winning pdf for this data points and then update the parameters for all the pdf
located in a neighborhood of the winning distribution (i.e. the pdf for which the maximum is
located on a point in a neighborhood of the data point chosen here).
6.9 Static Hopfield Network
In his 1982 seminal paper, John J. Hopfield [Hopfield, 1982] presented a novel type of associative
memory, consisting in a fully connected artificial neural network, later called the Hopfield Network.
This work was one of the first demonstrations that models of physical systems, such as neural
networks, could be used to solve computational problems. Later, follow-up on this research
showed that such systems could be implemented in hardware by combining standard
components such as capacitors and resistors.
The original Hopfield network, which we will refer to as the static Hopfield Network, consists of a
one-layer fully recurrent neural network. Subsequent development led Hopfield and coworkers to
develop different extensions. The major ones are the Bolzmann Machine, in which a parameter
(the temperature) can be varied so as to increase the capacity of the model, and the continuous
time Hopfield, whose neurons dynamics is continuous over time. In this class, we will consider
only the static and continuous time Hopfiel model. The interested reader can refer to [Barna &
Kaski, 1989; Ackley et al, 1985] for an introduction to the Boltzmann Machine.
The Hopfield network is created by supplying input data vectors, or pattern vectors,
corresponding to the different classes. These patterns are called class patterns. In an n-
dimensional data space the class patterns should have n binary components {1,-1}; that is, each
class pattern corresponds to a corner of a cube in an n-dimensional space. The network is then
used to classify distorted patterns into these classes. When a distorted pattern is presented to the
network, then it is associated with another pattern. If the network works properly, this associated
pattern is one of the class patterns. In some cases (when the different class patterns are
© A.G.Billard 2004 – Last Update March 2011