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Neural Network Hopfield model Kim, Il Joong Contents 1. Neural network: Introduction ① ② ③ 2. Hopfield model ① ② ③ 3. Definition & Application Network architectures Learning processes (Training) Summary of model Example Limitations Hopfield pattern recognition on a scale-free neural network Definition of Neural Network A massively parallel system made up of simple processing units and dense interconnections, which has a natural propensity for storing experiential knowledge and making it available for use. Interconnection strengths, known as synaptic weights, are used to store the acquired knowledge. => Learning process. Application of Neural Network Patterns-pattern mapping, pattern completion, pattern classification Image Analysis Speech Analysis & Generation Financial Analysis Diagnosis Automated Control Network architectures Single-layer feedforward network Network architectures Multilayer feedforward network Network architectures Recurrent network Learning processes (training) Error-correction learning Memory-based learning Hebbian learning Competitive learning Boltzmann learning Hebbian learning process If two neurons on either side of a synapse connection are activated simultaneously, then the strength of that synapse is increased. If two neurons on either side of a synapse are activated asynchronously, then the strength of that synapse is weakened or eliminated. Hopfield model Network architecture N processing units (binary) Fully(Infinitely) connected : N(N-1) connections Single-layer(no hidden layer) Recurrent(feedback) network : No self-feedback loof Hopfield model Learning process Let1 , 2 , 3 , , M denote a known set of N-dim. memories. 1 M W ( T M) N 1 Hopfield model Inputting and updating Let probe denote an unknown N-dimensional input vector. Update asynchronously (i.e., randomly and one at a time) according to the rule Hopfield model Convergence and Outputting Repeat updating until the state vector remains unchanged. Let X fixed denote the fixed point (stable state). Y X fixed Associated memories E 1 ji x j xi 2 j i i j 1 E j E j (n 1) E j (n) x j ji xi 2 i i j Memory vectors 1 , 2 , 3 , , M are states that corresponds to minimum E. Any input vector converges to the stored memory vector that is most similar or most accessible to the input. Hopfield model N=3 example Let (1,-1,1), (-1,1,-1) denote the stored memories. (M=2) 0 2 2 1 W 2 0 2 3 2 2 0 Limitations of Hopfield model ① The stored memories are not always stable. The signal-to-noise ratio: ② N M for large M. The quality of memory recall breaks down at M=0.14N There may be stable states that were not the stored memories. (Spurious states) Limitations of Hopfield model ③ Stable state may not be the state that is most similar to the input state. On a scale-free neural network Network architecture: the BA scale-free network A small core of m nodes. (fully connected) N (≫m) nodes are added. Total N + m processing units. Total Nm connections. (for 1≪m≪N) On a scale-free neural network Hopfield pattern recognition Stored P different patterns: i ( 1,2, , P) 1 Input pattern: 10% reversal of i ( =0.8) Output pattern: Si 1 The quality of recognition: overlap Sii1 N i On a scale-free neural network Small m : N=10000, m=2,3,5 On a scale-free neural network Large m : N+m=10000, P=10,100,1000 On a scale-free neural network Comparison with a fully connected network (m=N) For small m, low quality of recognition. For 1≪m≪N, good quality of recognition. Gain a factor N/m>>1 in the computer memory and time. A gradual decrease of quality of recognition. References A. S. Mikhailov, Foundations of Synergetics 1, Springer-Verlag Berlin Heidelberg (1990) John Hertz et al., Introduction to the theory of neural computation, Addison-Wesley (1991) Judith E. Dayhoff, Neural Network Architectures, Van Nostrand Reinhold (1990) S. Haykin, Neural Networks, Prentice-Hall (1999) D. Stauffer et al., http://xxx.lanl.gov/abs/cond-mat/0212601 (2002)
