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Feedback Networks and Associative Memories

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Content Introduction Discrete Hopfield NNs Continuous Hopfield NNs Associative Memories – Hopfield Memory – Bidirection Memory

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Feedback Networks and Associative Memories Introduction

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Feedforward/Feedback NNs Feedforward NNs – The connections between units do not form cycles. – Usually produce a response to an input quickly. – Most feedforward NNs can be trained using a wide variety of efficient algorithms. Feedback or recurrent NNs – There are cycles in the connections. – In some feedback NNs, each time an input is presented, the NN must iterate for a potentially long time before it produces a response. – Usually more difficult to train than feedforward NNs.

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Supervised-Learning NNs Feedforward NNs – Perceptron – Adaline, Madaline – Backpropagation (BP) – Artmap – Learning Vector Quantization (LVQ) – Probabilistic Neural Network (PNN) – General Regression Neural Network (GRNN) Feedback or recurrent NNs – Brain-State-in-a-Box (BSB) – Fuzzy Congitive Map (FCM) – Boltzmann Machine (BM) – Backpropagation through time (BPTT)

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Unsupervised-Learning NNs Feedforward NNs – Learning Matrix (LM) – Sparse Distributed Associative Memory (SDM) – Fuzzy Associative Memory (FAM) – Counterprogation (CPN) Feedback or Recurrent NNs – Binary Adaptive Resonance Theory (ART1) – Analog Adaptive Resonance Theory (ART2, ART2a) – Discrete Hopfield (DH) – Continuous Hopfield (CH) – Discrete Bidirectional Associative Memory (BAM) – Kohonen Self-organizing Map/Topology-preserving map (SOM/TPM)

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The Hopfield NNs In 1982, Hopfield, a Caltech physicist, mathematically tied together many of the ideas from previous research. A fully connected, symmetrically weighted network where each node functions both as input and output node. Used for – Associated memories – Combinatorial optimization

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Associative Memories An associative memory is a content-addressable structure that maps a set of input patterns to a set of output patterns. Two types of associative memory: autoassociative and heteroassociative. Auto-association – retrieves a previously stored pattern that most closely resembles the current pattern. Hetero-association – the retrieved pattern is, in general, different from the input pattern not only in content but possibly also in type and format.

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Associative Memories Auto-association A A Hetero-association Niagara Waterfall memory

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Optimization Problems Associate costs with energy functions in Hopfield Networks – Need to be in quadratic form Hopfield Network finds local, satisfactory soluions, doesnt choose solutions from a set. Local optimums, not global.

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Feedback Networks and Associative Memories Discrete Hopfield NNs

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The Discrete Hopfield NNs n n... v1v1 v2v2 v3v3 vnvn I1I1 I2I2 I3I3 InIn w 21 w 31 wn1wn1 w 12 w 32 wn2wn2 w 13 w 23 wn3wn3 w1nw1n w2nw2n w3nw3n

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The Discrete Hopfield NNs n n... v1v1 v2v2 v3v3 vnvn I1I1 I2I2 I3I3 InIn w 21 w 31 wn1wn1 w 12 w 32 wn2wn2 w 13 w 23 wn3wn3 w1nw1n w2nw2n w3nw3n w ij = w ij w ii = 0 w ij = w ij w ii = 0

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The Discrete Hopfield NNs n n... v1v1 v2v2 v3v3 vnvn I1I1 I2I2 I3I3 InIn w 21 w 31 wn1wn1 w 12 w 32 wn2wn2 w 13 w 23 wn3wn3 w1nw1n w2nw2n w3nw3n w ij = w ij w ii = 0 w ij = w ij w ii = 0

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State Update Rule Asynchronous mode Update rule Stable?

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Energy Function Fact: E is lower bounded (upper bounded). If E is monotonically decreasing (increasing), the system is stable.

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The Proof Suppose that at time t + 1, the k th neuron is selected for update.

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The Proof Suppose that at time t + 1, the k th neuron is selected for update. Their values are not changed at time t + 1.

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The Proof

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< vk(t)vk(t)v k (t+1)Hk(t)Hk(t) E E Stable

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Feedback Networks and Associative Memories Continuous Hopfield NNs

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The Neuron of Continuous Hopfield NNs uiui v i =a( u i ) 1 1 vivi u i =a 1 (v i ) gigi CiCi wi1wi1 wi2wi2 w in IiIi v1v1 v2v2 vnvn uiui vivi vivi

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The Dynamics gigi CiCi wi1wi1 wi2wi2 w in IiIi v1v1 v2v2 vnvn uiui vivi vivi GiGi

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The Continuous Hopfield NNs

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Stable?

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Equilibrium Points Consider the autonomous system: Equilibrium Points Satisfy

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The system is asymptotically stable if the following holds: Lyapunov Theorem There exists a positive- definite function E(y) s.t. Call E(y) as energy function.

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Lyapunov Energy Function

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g1g1 C1C1 I1I1 u1u1 g2g2 C2C2 I2I2 u2u2 g3g3 C3C3 I3I3 u3u3 gngn CnCn InIn unun v1v1 v2v2 v3v3 vnvn v1v1 v2v2 v3v3 vnvn w 21 wn1wn1 w 31 w 12 w 13 w1nw1n w 23 w2nw2n w 32 w3nw3n wn2wn2 wn3wn3... v u=a 1 (v) 1 1 v 1 1 uiui v i =a( u i ) 1 1

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Stability of Continuous Hopfield NNs Dynamics

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Stability of Continuous Hopfield NNs Dynamics > 0 v u=a 1 (v) 1 1

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Stability of Continuous Hopfield NNs Stable

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Feedback Networks and Associative Memories Associative Memories

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Associative Memories Also named content-addressable memory. Autoassociative Memory – Hopfield Memory Heteroassociative Memory – Bidirection Associative Memory (BAM)

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Associative Memories Stored Patterns Autoassociative Heteroassociative

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Feedback Networks and Associative Memories Associative Memories Hopfield Memory Bidirection Memory

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Hopfield Memory Neurons Fully connected 14,400 weights

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Example Stored Patterns Memory Association

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Example Stored Patterns Memory Association

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Example Stored Patterns Memory Association How to Store Patterns?

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The Storage Algorithm Suppose the set of stored pattern is of dimension n. =?

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The Storage Algorithm

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Analysis Suppose that x x i.

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Example

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Stable E=4 E=0 E= 4

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Example Stable E=4 E=0 E= 4

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Problems of Hopfield Memory Complement Memories Spurious stable states Capacity

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Capacity of Hopfield Memory The number of storable patterns w.r.t. the size of the network. Study methods: – Experiments – Probability – Information theory – Radius of attraction ( )

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Capacity of Hopfield Memory The number of storable patterns w.r.t. the size of the network. Hopfield (1982) demonstrated that the maximum number of patterns that can be stored in the Hopfield model of n nodes before the error in the retrieved pattern becomes severe is around 0.15n. The memory capacity of the Hopfield model can be increased as shown by Andrecut (1972).

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Radius of attraction ( 0 1/2 ) n False memories

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Feedback Networks and Associative Memories Associative Memories Hopfield Memory Bidirection Memory

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Bidirection Memory... y1y1 y2y2 ynyn x1x1 x2x2 xmxm X Layer Y Layer W=[w ij ] n m Forward Pass Forward Pass Backward Pass Backward Pass

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Bidirection Memory... y1y1 y2y2 ynyn x1x1 x2x2 xmxm X Layer Y Layer W=[w ij ] n m Forward Pass Forward Pass Backward Pass Backward Pass Stored Patterns ?

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The Storage Algorithm Stored Patterns

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Analysis Suppose x k is one of the stored vector. 0

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Energy Function

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Bidirection Memory Energy Function:

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