1. Synaptic Dynamics: Unsupervised Learning
Part Ⅱ
Wang Xiumei
2018/9/20

2. 1.Stochastic unsupervised learning and stochastic equilibrium;
2.Signal Hebbian Learning;
3.Competitive Learning.
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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.
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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.
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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 :
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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
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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[ ].
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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)
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9. The random competitive learning law
The random linear competitive learning law
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10. The learning vector quantization system.
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11. The self-organizing map system
The self-organizing map system equations:
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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.
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13. 2.Signal Hebbian Learning
⑴ Recency effects and forgetting;
⑵ Asymptotic correlation encoding;
⑶ Hebbian correlation decoding.
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14. Signal Hebbian Learning
The deterministic first-order
signal Hebbian learning law:
(4-132)
(4-133)
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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:
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16. Asymptotic correlation encoding
The synaptic matrix of long-term memory
traces asymptotically approaches the
bipolar correlation matrix :
X and Y denotes the bipolar signal vectors
and
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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)
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18. Hebbian correlation decoding
First we consider the bipolar correlation encoding of the M bipolar associations ,and turn bipolar associations into binary vector associations .
replace -1s with 0s
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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)
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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.
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21. Hebbian correlation decoding
Signal-noise decomposition:
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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 .
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23. Hebbian correlation decoding
We define the Hamming distance between binary vectors and
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24. Hebbian correlation decoding
[number of common bits]
- [number of different bits ]
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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.
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26. Hebbian correlation decoding
3) Suppose is far away from ,
. In the extreme case: ,
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27. Hebbian encoding method
binary vector
bipolar vector
sum contiguous correlation
-encoded associations:
Hebbian encoding method T
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28. Hebbian encoding method
Example(P144):
consider the three-step limit cycle:
convert bit vectors to bipolar vectors:
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29. Hebbian encoding method
Produce the asymmetric TAM matrix T:
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30. Hebbian encoding method
Passing the bit vectors through T in the
forward direction produces:
Produce the forward limit cycle:
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31. Competitive Learning The deterministic competitive learning law:
(4-165)
(4-166)
We see that the competitive learning law
uses the nonlinear forgetting term:
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32. Competitive Learning term . So the two laws differ in how they
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.
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33. 3.Competitive Learning. ⑴ Competition as Indication;
⑵ Competition as correlation
detection;
⑶ Asymptotic centroid estimation;
⑷ Competitive covariance estimation.
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34. Competition as indication
Centroid estimation requires that the
competitive signal approximate the
indicator function of the locally
sampled pattern class :
(4-168)
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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)
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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.
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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.
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38. Competition as correlation detection
Using equinorm property, we can get the
equivalent equalities(P147):
(4-174)
(4-178)
(4-179)
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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)
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40. Asymptotic centroid estimation
The simpler competitive law:
(4-181)
If we use the equilibrium condition:
(4-182)
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41. Asymptotic centroid estimation
Solving for the equilibrium synaptic vector:
(4-186)
It show that equals the centroid of .
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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.
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43. Competitive covariance estimation
Extend the competitive learning laws to
asymptotically estimate the local
conditional covariance matrices :
(4-187)
(4-189)
denotes the centriod.
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44. Competitive covariance estimation
The fundamental theorem of estimation
theory [Mendel 1987]:
(4-190)
is Borel-measurable random vector function
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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:
( )
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46. Competitive covariance estimation
denotes an appropriately decreasing sequence of
learning coefficients in(4-192).
If the ith neuron loses the metrical
competition
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47. Competitive covariance estimation
The algorithm(4-192) corresponds to the
stochastic differential equation:
(4-195)
(4-199)
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