CSE 5526: Introduction to Neural Networks Hopfield Network for Associative Memory

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1 CSE 5526: Introduction to Neural Networks Hopfield Network for Associative Memory Part VII 1

2 The basic task Store a set of fundamental memories {ξξ 1, ξξ 2,, ξξ MM } so that, when presented a new pattern xx, the system outputs one of the stored memories that is most similar to xx Such a system is called content-addressable memory Part VII 2

3 Architecture The Hopfield net consists of N McCulloch-Pitts neurons, recurrently connected among themselves xx 1 xx 2 xx NN ξξ 1 ξξ 2 ξξ NN Part VII 3

4 Definition Each neuron is defined as xx jj = φφ vv jj where vv jj = NN ii=1 ww jjjj xx ii + bb jj 1 if vv 0 and φφ vv = 1 otherwise Without loss of generality, let bb jj = 0 Part VII 4

5 Storage phase To store fundamental memories, the Hopfield model uses the outer-product rule, a form of Hebbian learning: MM ww jjjj = 1 NN ξξ μμ,jj ξξ μμ,ii μμ=1 Hence ww jjjj = ww iijj, i.e., ww = ww TT, so the weight matrix is symmetric Part VII 5

6 Retrieval phase The Hopfield model sets the initial state of the net to the input pattern. It adopts asynchronous (serial) update, which updates one neuron at a time Part VII 6

7 Hamming distance Hamming distance between two binary/bipolar patterns is the number of differing bits Example (see blackboard) Part VII 7

8 One memory case Let the input xx be the same as the single memory ξξ xx jj = φφ ww jjii ii xx ii = φφ 1 NN ξξ jjξξ ii ξξ ii = φφ ξξ jj = ξξ jj ii Therefore the memory is stable Part VII 8

9 One memory case (cont.) Actually for any input pattern, as long as the Hamming distance between xx and ξξ is less than NN 2, the net converges to ξξ. Otherwise it converges to ξξ Think about it Part VII 9

10 Multiple memories The stability (alignment) condition for any memory ξξ θθ is φφ vv jj = ξξ θθ,jj where vv jj = ww jjjj ξξ θθ,ii = 1 NN ξξ μμ,jjξξ μμ,ii ξξ θθ,ii ii ii μμ = ξξ θθ,jj + 1 NN ξξ μμ,jjξξ μμ,ii ξξ θθ,ii ii μμ θθ crosstalk Part VII 10

11 Multiple memories (cont.) If crosstalk < N, the memory system is stable. In general, fewer memories are more likely stable Example 2 in book (see blackboard) Part VII 11

12 Example (cont.) Part VII 12

13 Memory capacity Define CC jj θθ = ξξ θθ,jj ξξ μμ,jj ξξ μμ,ii ξξ θθ,ii ii μμ θθ CC θθ jj < 0 stable 0 CC θθ jj < NN stable CC θθ jj > NN unstable What if CC jj θθ = NN? Part VII 13

14 Memory capacity (cont.) Consider random memories where each element takes +1 or -1 with equal probability (prob.). We measure PPerror = Prob(CC jj θθ > NN) To compute capacity M max, decide on an error criterion For random patterns, CC jj θθ is proportional to a sum of NN(MM 1) random numbers of +1 or -1. For large NNMM, it can be approximated by a Gaussian distribution (central limit theorem) with zero mean and variance σσ 2 = NNMM Part VII 14

15 Memory capacity (cont.) So PPerror = 1 2πσσ xx2 exp ( 2σσ 2) dxx NN = πσσ exp 0 NN xx2 2σσ 2 dxx (xx = 2σσσσ) = π NN/(2MM) exp μμ2 0 dμμ error function Part VII 15

16 Memory capacity (cont.) Error probability plot PP(CC jj θθ ) N x Part VII 16

17 Memory capacity (cont.) PPerror MMmax/NN So PPerror < 0.01 MMmax = 0.185NN, an upper bound Part VII 17

18 Memory capacity (cont.) What if 1% flips occur? Avalanche effect? Real upper bound: 0.138N to prevent the avalanche effect Part VII 18

19 Memory capacity (cont.) The above analysis is for one bit. If we want perfect retrieval for ξξ θθ with probability of 0.99: (1 PPerror) NN > 0.99 Approximately PPerror < 0.01 NN For this case MMmax = NN 2log NN Part VII 19

20 Memory capacity (cont.) But real patterns are not random (they could be encoded) and the capacity is worse for correlated patterns At one favorable extreme, if memories are orthogonal ξξ μμ,ii ξξ θθ,ii = 0 for θθ μμ ii then CC jj θθ = 0 and MMmax = NN This is the maximum number of orthogonal patterns Part VII 20

21 Memory capacity (cont.) But in reality a useful system stores a little less; otherwise the memory is useless as it does not evolve Part VII 21

22 Coding illustration Part VII 22

23 Energy function (Lyapunov function) The existence of an energy (Lyapunov) function for a dynamical system ensures its stability The energy function for the Hopfield net is EE(xx) = 1 2 ww jjjjxx ii xx jj ii jj Theorem: Given symmetric weights, ww jjjj = ww iijj, the energy function does not increase as the Hopfield net evolves Part VII 23

24 Energy function (cont.) Let xx jj be the new value of xx jj after an update xx jj = φφ ww jjjj xx ii ii If xx jj = xx jj, EE remains the same Part VII 24

25 In the other case, xx jj = xx jj : Energy function (cont.) EE xx jj EE xx jj = 1 2 ww jjjjxx ii xx jj ww jjjjxx ii xx jj ii jj ii jj since ww jjjj = MM NN = 1 2 ww jjjjxx ii xx jj ww jjjjxx ii xx jj ii jj jj ii jj jj since ww iijj = ww jjjj = xx jj ww jjjj xx ii ii jj + xx jj ww jjjj xx ii ii jj = 2xx jj ww jjjj xx ii ii jj = 2xx jj ww jjjj xx ii 2ww jjjj < 0 ii Part VII different signs 25

26 Energy function (cont.) Thus, EE(xx) decreases every time xx jj flips. Since EE is bounded, the Hopfield net is always stable Remarks: Useful concepts from dynamical systems: attractors, basins of attraction, energy (Lyapunov) surface or landscape Part VII 26

27 Energy contour map Part VII 27

28 2-D energy landscape Part VII 28

29 Memory recall illustration Part VII 29

30 Remarks (cont.) Bipolar neurons can be extended to continuous-valued neurons by using hyperbolic tangent activation function, and discrete update can be extended to continuous-time dynamics (good for analog VLSI implementation) The concept of energy minimization has been applied to optimization problems (neural optimization) Part VII 30

31 Spurious states Not all local minima (stable states) correspond to fundamental memories. Typically, ξξ μμ, linear combination of odd number of memories, or other uncorrelated patterns, can be attractors Such attractors are called spurious states Part VII 31

32 Spurious states (cont.) Spurious states tend to have smaller basins and occur higher on the energy surface local minima for spurious states Part VII 32

33 Kinds of associative memory Autoassociative (e.g. Hopfiled net) Heteroassociative: store pairs of < xx μμ, yy μμ > explicitly xx matrix memory holographic memory yy Part VII 33

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