Hopfield Network Recurrent Netorks

Transcription

1 Hopfield Network Recurrent Netorks w 2 w n E.P. P.E. y (k) Auto-Associative Memory: Given an initial n bit pattern returns the closest stored (associated) pattern. No P.E. self-feedback! w 2 w n2 E.P.2 P.E.2 y 2 (k) ( k) Dynamics : s n i w i y ( k) i ( ) ( s ) ( k+ ) k f y w n w 2n E.P.n P.E.n y n (k) Hopfield Network with n Processing Elements Network Initialization: Binary f ( s Output Vector: activation ) hold if if s s () y x y function : ( ) [ y ] ( k) k i > L < L previous value, if s L Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 84

2 Hopfield Network... w 2 w n E.P. P.E. y (k) - Fast training and fast data recovery - IIR system with no input (only I.C.) - Guaranteed stability - Good for VLSI implementation w 2 w n2 E.P.2 P.E.2 y 2 (k) -Operating Forms (firing order) Asynchronous Synchronous Sequential w n w 2n E.P.n P.E.n y n (k) Initial Condition End Value (stable) Hopfield Network with n Processing Elements Possible Hopfield Network states (8) with Processing Elements (Illustration of a typical recovery state evolution. From I.C. to E.V.) Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 85

3 Hopfield Network... Learning: The patterns to be stored in the associative memory are chosen a priori. m distinct patterns. Each of the form: A p p p p [ a a a ], with a or. ( L, ) p 2 K n i usually w i m p (2a p i )(2a p ) p a i ( 2 ) Obs: converts / to /+ p i w i is incremented by if a a otherwise it is decremented Procedure is repeated for each i, for every A p. Learning is analogous to reinforcement learning p Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 86

4 Hopfield Network - Example Symbol Training Vector L A [ ] T A 2 [ ] a a 2 a a 4 a 5 a 6 a 7 a 8 a 9 Patterns to be stored as x matrices: Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 87 + A [ ] W m p p p i i a a w ) )(2 2 ( Weigth Matrix

5 Hopfield Network - Example ] [ () y x New pattern presented to the trained network: Fired P.E. P.E. Sum P.E. Output New output vector Sequential operation of the network: Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 88 ] [ W y x < >, ) ( :, s if value previous hold s if s if s f L function activation Binary Remember Convergence to L Pattern

6 Hopfield Network ava demos Demonstrations available in the www, e.g.: Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 89

7 Hopfield N. final considerations Stability proof Cohen and Grossberg, 98. W symmetric with zero diagonal Energy funcition always decreases. E 2 i w i y i y y L E - Energia Energy da rede of the network Spurious Padrão Pattern espúrio Initial Valor Inicial State Hopfield Network Limitations: Not necessarily the closest pattern is returned. Differences between patterns. Not all patterns have equal emphasis (size of attraction basins). Spurious patterns, i.e., patterns evoked that are not part of the stored set. Maximum number of stored patterns is limited. m,5n / log n, m patterns, n bits network Padrão Recovered recuperado Pattern Stored Padrões Patterns armazenados Estados States Typical Energy and Patterns illustration for Hopfield Newtworks Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 9

8 Radial Basis Functions - Moody & Darken, 989,... - Function Approximators - Inspiration: sensoricc overlapped reception fields in the cortex - Localized activity of the processing elements a i e µ x i i 2 σ i Gaussian (Average, Variance) a a i e p w i i b ml: w weigth b bias.4 Weighted Sum of Radial Basis Transfer Functions.2 pw pw2 pwn Output a Input p p Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 9

9 Radial Basis Functions... a a radbas( dist( w, p)* b) X: -.8 Y: w-p MatLab Implementation [net,tr] newrb(p,t,goal,spread,mn) P - RxQ matrix, Q input vectors ( pattern ), T - SxQ matrix, Q obective vectors ( target ), GOAL - desired mean square error, default., SPREAD - radial basis function spread, default., MN - Maximum number of neurons, default is Q. Output a Weighted Sum of Radial Basis Transfer Functions Input p Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 92

10 Radial Basis Functions... Learning: add neurons incrementally - Where? To obtain the largest quadratic errror reductions at each step min e spread to low - Good fitting (at the training points)! - Bad interpolation! spread to high - Good fitting at low frequencies - Good interpolation in some ranges! Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 9

11 Radial Basis Functions... Heuristics: x i+ xi < SPREAD < xmax xmin Target function 26 Amostras samplesda Função Function Função Aprox. Aprox. with 2 RBF RBF - 2 neuros neurônios spread OK - Good fitting! - Good interpolation! Bad extrapolation (is a very difficult task) Pattern Conclusions - Faster training faster, but uses more neurons than MLP. - Incremental Training, new points can be learned without losing prior knowledge. -You can use a priori knowledge to locate neurons (which is not possible in a MLP). - Fixed spread Incremental traning suboptimal solution!! Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 94

12 Comparison RBF x MLP RBF comparison - MSE over ti:.:,yiexp(-.*ti).*sin(.*ti.*ti)) samples (2) function MSE 5 RBFs.62 RBFs.25 5 RBFs.28 2 RBFs MLP comparison - MSE over ti:.:,yiexp(-.*ti).*sin(.*ti.*ti)) samples (2) function MSE 5 MLPs.672 MLPs MLPs MLPs RBF more neurons better fitting best solution newrbe (exact fitting!) MLP too much neurons worse fitting (bad interpolation) Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 95

13 Unsupervised Learning Competitive Layer Find vector codes that describe the data distribution No known desired Code Vectors They should reflect the Inner statistical distribution of the process Used in Data Compression Example: Code symbols that will be transmitted over a communication channel. For the comprehension the variability of the signal is considered as noise, and so, discarded. Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 96

14 Competitive Layer o code vectors training vectors x test vectors Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 97

15 Borders (Voronoi Diagram) o code vectors + training vectors x test vectors Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 98

16 Ex. Classification Borders o code vectors + training vectors x test vectors Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 99

17 Bias Bias adustment to to help weak neurons Bias (only Euclidean distance) Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic

18 Learning Vector Quantization No known desired Code Vectors Data points belong to Classes Code Vectors should reflect the Inner statistical distribution of the process Class A Class B Input Vectors Current input Class C Code vectors Class(X)Class(Wc) LVQ, LVQ2., LVQ, OLVQ Enhanced Algorithms -Dead neurons -Neighborhood definition Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic

19 Self Organizing Maps Kohonen, 982 Unsupervised learning One active layer with neighborhood constrains Code Vectors should reflect the Inner statistical distribution of the process Triangular distribution.6 Weight Vectors SOM Map W(i,2) W(i,) Weird unsuccessful training Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic 2

20 ANN General Characteristics Positive Learning Parallelism Distributed knowledge Fault Tolerant Associative Memory Robust against Noise No exhaustive modelling To obtain successful ANN a good process knowledge is recommended in order to design experiments that produce useful data sets! Negative Knowledge acquisition only by learning ( E.g., Wich topology is best suit?) Introspection is not possible ( What is the contribution of this neuron?) The logical inference is hard to obtain ( Why this output for this situation? ) Learning is slow Very sensitive to initial conditions There is no free lunch! Laboratório de Automação e Robótica - A. Bauchspiess Soft Computing - Neural Networks and Fuzzy Logic