Hopfield NetworksPresentation By Utkarsh TrivediY8544

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Topics Covered

•What is Hopfield Network•Some interesting facts•Major Applications•Mathematical model of HN’s•Learning HNs through examples

Hopfield Network

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What is Hopfield Network ??

• According to Wikipedia, Hopfield net is a form of recurrent artificial neural network invented by John Hopfield. Hopfield nets serve as content-addressable memory systems with binary threshold units. They are guaranteed to converge to a local minimum, but convergence to one of the stored patterns is not guaranteed.

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What are HN (informally)Hopfield Network

•These are single layered recurrent networks

•All the neurons in the network are fedback from all other neurons in the network

•The states of neuron is either +1 or -1 instead of (1 and 0) in order to work correctly.

•No of the input nodes should always be equal to no of output nodes

•Following figure shows a Hopfield network with four nodes

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Some Interesting Facts ….• The recall pattern of Hopfield network is similar

to our recall mechanism.• Both are based on content addressable memory• If some of the neurons of network are destroyed

the performance is degraded but some network capabilities may be retained even with major network damage. Just like our brains

Hopfield Network

Did you know that

we are similar

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Major Applications• Recalling or Reconstructing corrupted patterns• Large-scale computational intelligence systems• Handwriting Recognition Software• Practical applications of HNs are limited because number

of training patterns can be at most about 14% the number of nodes in the network.

• If the network is overloaded -- trained with more than the maximum acceptable number of attractors -- then it won't converge to clearly defined attractors.

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Mathematical Modeling of HN’s

Hopfield Network

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Mathematical Modeling of HN’s

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•Consider signum function to be neuron’s activation function.

• i.e. vi = +1 if hi>0

vi = -1 if hi<0

Mathematical Modeling of HN’s

Hopfield Network

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Mathematical Modeling of HN’s

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Power Of Hopfield Networks

We want to understand how to achieve this kind of performance form simple Hopfield networks

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Learning HNs through simple example

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There are various ways to train these kinds of networks like back propagation algorithm , recurrent learning algorithm, genetic algorithm but there is one very simple algorithm to train these simple networks called ‘One shot method’. I will be using this algorithm in order to train the network.

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Learning HNs through simple example

•Lets train this network for following patterns

• Pattern 1:- ie Oa(1)=-1,Ob(1)=-1,Oc(1)=1

• Pattern 2:- ie Oa(2)=1,Ob(2)=-1,Oc(3)=-1

• Pattern 3:- ie Oa(3)=-1,Ob(3)=1,Oc(3)=1

w1,1 = 0w1,2 = OA(1) × OB(1) + OA(2) × OB(2) + OA(3) × OB(3) = (-1) × (-1) + 1 × (-1) + (-1) × 1 = 1w1,3 = OA(1) × OC(1) + OA(2) × OC(2) + OA(3) × OC(3) = (-1) × 1 + 1 × (-1) + (-1) × 1 = -3w2,2 = 0w2,1 = OB(1) × OA(1) + OB(2) × OA(2) + OB(3) × OA(3) = (-1) × (-1) + (-1) × 1 + 1 × (-1) = -1w2,3 = OB(1) × OC(1) + OB(2) × OC(2) + OB(3) × OC(3) = (-1) × 1 + (-1) × (-1) + 1 × 1 = 1w3,3 = 0w3,1 = OC(1) × OA(1) + OC(2) × OA(2) + OC(3) × OA(3) = 1 × (-1) + (-1) × 1 + 1 × (-1) = -3w3,2 = OC(1) × OB(1) + OC(2) × OB(2) + OC(3) × OB(3) = 1 × (-1) + (-1) × (-1) + 1 × 1 = 1

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Learning HNs through example• Moving onto little more complex problem described in Haykin’s

Neural Network Book• They book used N=120 neuron and trained network with 120 pixel

images where each pixel was represented by one neuron.• Following 8 patterns were used to train neural network.

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Learning HNs through example

•In order to recognizing power of HNs•For this they need corrupted image. They

flipped the value of each pixel with p=0.25.

•Using these corrupted images trained HN was run. And after certain number of iteration the output images converged to one of the learned pattern.

•Next slides shows the results that they obtained

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Learning HNs through example

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Learning HNs through example

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Flow Chart summarizing overall process

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Shortcomings of HNs• Training patterns can be at most about 14% the

number of nodes in the network.• If more patterns are used then

the stored patterns become unstable; spurious stable states appear (i.e., stable states

which do not correspond with stored patterns).• Can sometimes misinterpret the corrupted

pattern.

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Shortcomings of HNs

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References• Zurada :- Introduction to Artificial Neural

Systems• Haykins :- Neural Networks, A Comprehensive

Foundation• J. J. Hopfield, "Neural networks and physical

systems with emergent collective computational abilities",1982

• R. Rojas: Neural Networks, Springer-Verlag, Berlin, 1996

• Wikipedia :-http://en.wikipedia.org/wiki/Hopfield_networks

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Thank you …..

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Questions…• What is the major difference between HN and fully

recurrent networks?• What is content addressable memory and how is it

different from RAM?• What will happen if we train HN for only one

pattern?• If we train a HN with a pattern will it be

automatically trained for its inverse ?• Why can’t we increase number of nodes in

network in order to overcome its shortcomings? (ignore the increased computation complexity or

time)