Mathematical Foundation of Discrete time Hopfield Networks
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Transcript of Mathematical Foundation of Discrete time Hopfield Networks
A Seminar Presentation for the degree of
Master of Technology in
Computer Science and Engineering
PRESENTED BY:-AKHIL UPADHYAYM-TECH 3rd SEM CSEROLL NO.- 121140002
SUBMITED TO:-MR. ROHIT MIRIH.O.D. OF COMPUTER SCIENCE DEPARTMENT
Mathematical Foundation of Discrete time Hopfield Networks
INTRODUCTION A Hopfield Networks is a form of recurrent artificial neural Network popularized by John Hopfield in 1982, but described earlier by Little in 1974
Hopfield has developed a number of neural Networks based on fixed weights and adaptive activations
These Networks can serve as associative memory Networks and can be used to solve constraint satisfaction problems such as the "Travelling Salesman Problem (Cont..)
Two types:
1. Discrete Hopfield Network.
2. Continuous Hopfield Network.
Discrete Hopfield Network
Hopfield has proposed two basic models of associative memories (Hopfield 1982, 1984).
(Cont..)
Discrete Hopfield NetworkThe first of these is a ‘DISCRETE MODEL’ while the second is a ‘CONTINUOUS’ version of the same.
The terms ‘DISCRETE’ or ‘CONTINUOUS’ refer to the nature of the state variables and time, in these models.
In the discrete Hopfield network, each neuron has a binary state
{1,-1} The state of the network with N neurons is represented
by the vector
(Cont..)
Discrete Hopfield Network
V={ The network is fully-connected, i.e., each neuron
connected to all others.
The weight from j’th neuron to i’th neuron is given by, and weight matrix is given as
W={} Since the network has loops, computations are dynamic
and the network state evolves through time, which is a discrete variable.
(Cont..)
Discrete Hopfield Network
Hopfield net differ from iterative auto associative net in 2
things.
1. Only one unit updates its activation
at a time (based on the signal it receives from each other
unit)
2. Each unit continues to receive an
external signal in addition to the signal from the other units
in the net.
(Cont..)
Surprise
The asynchronous updating of the units allows a function,
known as an energy function, to be found for the net.
The existence of such a function enables us to prove that the
net will converge to a stable set of activations, rather than
oscillating.
The original formulation of the discrete Hopfield net showed
the usefulness of the net as content-addressable memory.
(Cont..)
Discrete Hopfield Network
(Cont..)
Discrete Hopfield Network
Algorithm There are several versions of the discrete Hopfield net.
Binary Input Vectors
To store a set of binary patterns s ( p ) ,
p = 1 , . . . , P, where
))().....().....(()( 1 pspspsps ni
(Cont..)
Discrete Hopfield Network
The weight matrix W = is given by}{ ijw
]12][12[ )()( pjp
piij sswji for
and
.0iiw
(Cont..)
Discrete Hopfield Network Bipolar Inputs
To store a set of binary patterns s ( p ) ,
p = 1 , . . . , P, where
))().....().....(()( 1 pspspsps ni
The weight matrix W = is given by,}{ ijw
)()( pjp
piij ssw ji for
and0iiw (Cont..)
PROPERTIES OF HOPFIELD NETWORK
A recurrent network with all nodes connected to all other nodes. Nodes have binary outputs (either 0,1 or -1,1). Weights between the nodes are symmetric . No connection from a node to itself is allowed. Nodes are updated asynchronously ( i.e. nodes are selected at random). The network has no hidden nodes or layer.
(Cont..)
Discrete Hopfield Network
Applications:-A binary Hopfield net can be used to determine whether an input
vector is a "known” or an "unknown" vector.
The net recognizes a "known" vector by producing a pattern of
activation on the units of the net that is the same as the vector
stored in the net.
If the input vector is an "unknown" vector, the activation vectors
produced as the net iterates will converge to an activation vector
that is not one of the stored patterns.
Thank You