Hopfield Networks

1. Hopfield Networks Presentation By Utkarsh Trivedi Y8544

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

3. Hopfield Network 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.

4. Hopfield Network What are HN (informally) • 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

5. Hopfield Network 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 Did you know that we are similar

6. Hopfield Network 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.

7. Hopfield Network Mathematical Modeling of HN’s

8. Hopfield Network Mathematical Modeling of HN’s

9. Hopfield Network Mathematical Modeling of HN’s • Consider signum function to be neuron’s activation function. • i.e. vi = +1 if hi>0 vi = -1 if hi<0

10. Hopfield Network Mathematical Modeling of HN’s Liapunov Energy function :-

11. Hopfield Network Power Of Hopfield Networks We want to understand how to achieve this kind of performance form simple Hopfield networks

12. Hopfield Network Learning HNs through simple example 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. W1,1 Oa W2,1 W1,2 W3,1 W1,3 W2,2 Ob W3,2 W2,3 Oc W3,3

13. Hopfield Network Learning HNs through simple example • Lets train this network for following patterns • Pattern 1:- ieOa(1)=-1,Ob(1)=-1,Oc(1)=1 • Pattern 2:- ieOa(2)=1,Ob(2)=-1,Oc(3)=-1 • Pattern 3:- ieOa(3)=-1,Ob(3)=1,Oc(3)=1 w1,1 = 0 w1,2 = OA(1) × OB(1) + OA(2) × OB(2) + OA(3) × OB(3) = (-1) × (-1) + 1 × (-1) + (-1) × 1 = 1 w1,3 = OA(1) × OC(1) + OA(2) × OC(2) + OA(3) × OC(3) = (-1) × 1 + 1 × (-1) + (-1) × 1 = -3 w2,2 = 0 w2,1 = OB(1) × OA(1) + OB(2) × OA(2) + OB(3) × OA(3) = (-1) × (-1) + (-1) × 1 + 1 × (-1) = -1 w2,3 = OB(1) × OC(1) + OB(2) × OC(2) + OB(3) × OC(3) = (-1) × 1 + (-1) × (-1) + 1 × 1 = 1 w3,3 = 0 w3,1 = OC(1) × OA(1) + OC(2) × OA(2) + OC(3) × OA(3) = 1 × (-1) + (-1) × 1 + 1 × (-1) = -3 w3,2 = OC(1) × OB(1) + OC(2) × OB(2) + OC(3) × OB(3) = 1 × (-1) + (-1) × (-1) + 1 × 1 = 1

14. Hopfield Network 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.

15. Hopfield Network 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

16. Hopfield Network Learning HNs through example

17. Hopfield Network Learning HNs through example

18. Hopfield Network Flow Chart summarizing overall process

19. Hopfield Network 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.

20. Hopfield Network Shortcomings of HNs

21. Hopfield Network 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

22. Hopfield Network Thank you …..

23. Hopfield Network 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)