Synaptic Dynamics: Unsupervised Learning

1. Synaptic Dynamics:Unsupervised Learning Part Ⅱ Wang Xiumei

2. 1.Stochastic unsupervised learning and stochastic equilibrium; 2.Signal Hebbian Learning; 3.Competitive Learning.

3. 1.Stochastic unsupervised learning and stochastic equilibrium ⑴ The noisy random unsupervised learning law; ⑵ Stochastic equilibrium; ⑶ The random competitive learning law; ⑷ The learning vector quantization system.

4. The noisy random unsupervised learning law The random-signal Hebbian learning law: (4-92) denotes a Browian-motion diffusion process, each term in (4-92)demotes a separate random process.

5. The noisy random unsupervised learning law • Using noise relationship: we can rewrite (4-92): (4-93) We assume the zero-mean, Gaussian white-noise process ,and use equation :

6. The noisy random unsupervised learning law We can get a noisy random unsupervised learning law (4-94) Lemma: (4-95) is finite variance. proof: P132

7. The noisy random unsupervised learning law The lemma implies two points: 1, stochastic synapses vibrate in equilibrium, and they vibrate at least as much as the driving noise process vibrates; 2,the synaptic vector changes or vibrate at every instant t, and equals a constant value. wanders in a brownian motion about the constant value E[ ].

8. Stochastic equilibrium When synaptic vector stops moving, synaptic equilibrium occurs in “steady state”, (4-101) synaptic vector reaches synaptic equilibrium when only the random noise vector change : (4-103)

9. The random competitive learning law The random competitive learning law The random linear competitive learning law

10. The learning vector quantization system.

11. The self-organizing map system • The self-organizing map system equations:

12. The self-organizing map system The self-organizing map is a unsupervised clustering algorithm. Compared with traditional clustering algorithms, its centroid can be mapped a curve or plain, and it remains topological structure.

13. 2.Signal HebbianLearning ⑴Recency effects and forgetting; ⑵Asymptotic correlation encoding; ⑶Hebbian correlation decoding.

14. Signal Hebbian Learning The deterministic first-order signal Hebbian learning law: (4-132) (4-133)

15. Recency effects and forgetting Hebbian synapses learn an exponentially weighted average of sampled patterns. the forgetting term is . The simplest local unsupervised learning law:

16. Asymptotic correlation encoding The synaptic matrix of long-term memory tracesasymptotically approaches the bipolar correlation matrix : X and Y denotes the bipolar signal vectors and .

17. Asymptotic correlation encoding In practice we use a diagonal fading-memory exponential matrix W compensates for the inherent exponential decay of learned information: (4-142)

18. Hebbian correlation decoding First we consider the bipolar correlation encoding of theM bipolar associations ,and turn bipolar associations into binary vector associations . replace -1s with 0s

19. Hebbian correlation decoding The Hebbian encoding of the bipolar associations corresponds to the weighted Hebbian encoding scheme if the weight matrix W equals the (4-143)

20. Hebbian correlation decoding We use the Hebbian synaptic M for bidirectional processing of and neuronal signals, and pass neural signal through M in the forward direction, in the backward direction.

21. Hebbian correlation decoding Signal-noise decomposition:

22. Hebbian correlation decoding Correction coefficients : (4-149) They can make each vector resemble in sign as much as possible. The same correction property holds in the backward direction .

23. Hebbian correlation decoding We definethe Hamming distance between binary vectors and

24. Hebbian correlation decoding [number of common bits] -[number of different bits ]

25. Hebbian correlation decoding • Suppose binary vector is close to , Then ,geometrically, the two patterns are less than half their space away from each other, So . In the extreme case ;so . • The rare case that result in , and the correction coefficients should be discarded.

26. Hebbian correlation decoding 3) Suppose is far away from , . In the extreme case: , .

27. binary vector bipolar vector sum contiguous correlation -encoded associations: Hebbian encoding method T

28. Hebbian encoding method Example(P144): consider the three-step limit cycle: convert bit vectors to bipolar vectors:

29. Hebbian encoding method Produce the asymmetric TAM matrix T:

30. Hebbian encoding method Passing the bit vectors through T in the forward direction produces: Produce the forward limit cycle:

31. Competitive Learning The deterministic competitive learning law: (4-165) (4-166) We see that the competitive learning law uses the nonlinear forgetting term: .

32. Competitive Learning Heb learning law uses the linear forgetting term . So the two laws differ in how they forget, not in how they learn. In both cases when -when the jth competing neuron wins-the synaptic value encodes the forcing signal and encodes it exponentially quickly.

33. 3.Competitive Learning. ⑴ Competition as Indication; ⑵ Competition as correlation detection; ⑶ Asymptotic centroid estimation; ⑷ Competitive covariance estimation.

34. Competition as indication Centroid estimation requires that the competitive signal approximate the indicator function of the locally sampled pattern class : (4-168)

35. Competition as indication If sample pattern X comes from region , the jth competing neuron in should win, and all other competing neurons should Lose. In practice we usually use the random linear competitive learning law and a simple additive model. (4-169)

36. Competition as indication the inhibitive-feedback term equals the additive sum of synapse-weighted signal: (4-170) if the jth neuron wins, and to if instead the kth neuron wins.

37. Competition as correlation detection The metrical indicator function: (4-171) If the input vector X is closer to synaptic vector than to all other stored synaptic vectors, the jth competing neuron will win.

38. Competition as correlation detection Using equinorm property, we can get the equivalent equalities(P147): (4-174) (4-178) (4-179)

39. Competition as correlation detection From the above equality, we can get: The jth Competing neuron wins iff the input signal or pattern correlates maximally with . The cosine law: (4-180)

40. Asymptotic centroid estimation The simpler competitive law: (4-181) If we use the equilibrium condition: (4-182)

41. Asymptotic centroid estimation Solving for the equilibrium synaptic vector: (4-186) It show that equals the centroid of .

42. Competitive covariance estimation Centroids provides a first-order Estimate of how the unknown probability Density function behaves in the regions , and local covariances provide a second-order description.

43. Competitive covariance estimation Extend the competitive learning laws to asymptotically estimate the local conditional covariance matrices : (4-187) (4-189) denotes the centriod.

44. Competitive covariance estimation The fundamental theorem of estimation theory [Mendel 1987]: (4-190) is Borel-measurable random vector function

45. Competitive covariance estimation At each iteration we estimate the unknown centroid as the current synaptic vector ,In this sense becomes an error conditional covariance matrix . the stochastic difference-equation algorithm: (4-191-192)

46. Competitive covariance estimation denotes an appropriately decreasing sequence of learning coefficients in(4-192). If the ith neuron loses the metrical competition

47. Competitive covariance estimation The algorithm(4-192) corresponds to the stochastic differential equation: (4-195) (4-199)