synaptic dynamics: unsupervised learning
DESCRIPTION
Synaptic Dynamics: Unsupervised Learning. Part Ⅱ Wang Xiumei. 1. Stochastic unsupervised learning and stochastic equilibrium; 2. Signal Hebbian Learning; 3. Competitive Learning. 1.Stochastic unsupervised learning and stochastic equilibrium. ⑴ The noisy random unsupervised - PowerPoint PPT PresentationTRANSCRIPT
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Synaptic Dynamics:Synaptic Dynamics:Unsupervised Unsupervised
LearningLearningPart ⅡPart Ⅱ
Wang XiumeiWang Xiumei
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1.Stochastic unsupervised learning and stochastic equilibrium;2.Signal Hebbian Learning;
3.Competitive Learning.
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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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The noisy random unsupervised learning
lawThe random-signal Hebbian learning law: (4-92) denotes a Browian-motion diffusion process, each term in (4-92)demotes a separate random process.
( ) ( )ij ij i i i i ijdm m dt S x S y dt dB
{ ( )}ijB t
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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 :
( ) ( )ij ij i i j j ijm m S x S y n
dB ndt
{ ( )}ijn t
( , , ) ( ) ( )ij ij i i j jf x y M m S x S y
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The noisy random unsupervised learning
lawWe can get a noisy random unsupervised
learning law (4-94)Lemma: (4-95)
is finite variance. proof: P132
, ,ij ij ijm f X Y M n
2ij ijE m
ij
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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[ ].
jm
jmjm
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Stochastic equilibriumWhen synaptic vector stops moving,synaptic equilibrium occurs in “steady state”, (4-101)synaptic vector reaches synaptic equilibrium when only the random noisevector change : (4-103)
jm
jm
jm
0jm
j jm njn
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The random competitive learning
lawThe random competitive learning law
The random linear competitive learning law j j j j jm S y S X m n
j j j j jm S y X m n
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The learning vector quantization
system.
1
1
1
j j k k j k j
j j k k j k j
i i
m k m k c X m k X D
m k m k c X m k X D
m k m k i j
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The self-organizing map system
• The self-organizing map system equations:
1
1j j k k j
i i
m k m k C X m k
m k m k i j
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The self-organizing map system
The self-organizing map is a unsupervisedclustering algorithm.Compared with traditional clusteringalgorithms, its centroid can be mapped a curve or plain, and it remains topologicalstructure.
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2.Signal Hebbian Learning⑴ Recency effects and forgetting;⑵ Asymptotic correlation encoding;⑶ Hebbian correlation decoding.
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Signal Hebbian LearningThe deterministic first-ordersignal Hebbian learning law: (4-132)
(4-133)
ij ij i i j jm m t S x t S y t
0
0tt s t
ij ij i jm t m e S s S s e ds
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Recency effects and forgettingHebbian synapses learn an exponentially weighted average of sampled patterns.the forgetting term is .The simplest local unsupervised learninglaw:
ijm
ij ijm m
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Asymptotic correlation encoding
The synaptic matrix of long-term memory
traces asymptotically approaches thebipolar correlation matrix :
X and Y denotes the bipolar signal vectors and .
TK KX Y
TK KM X Y
Mijm
( )S x ( )S y
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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)1
mT T
k k kk
X WY w X Y
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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
,i iX Y
,i iX Y
,i iA B
,i iA B ,i iX Y
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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)1
mTi i
i
M X Y
mmI
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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.
TM
XF YF
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Hebbian correlation decodingSignal-noise decomposition:
m
T Ti i i i i j j
j i
X M X X Y X X Y
m
Ti i j j
j i
nY X X Y
ij j
j
c Y
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Hebbian correlation decodingCorrection coefficients : (4-149) They can make each vector resemble
in sign as much as possible. The same correction property holds in the backward direction .
T Tij i j j i jic X X X X c
ij jc Y
ijc
jY
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Hebbian correlation decoding
We define the Hamming distance between binary vectors and iA jA
1
,n
k ki j i j
k
H A A a a
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Hebbian correlation decoding [number of common bits] - [number of different bits ]
Tij i jc X X
, ,i j i jn H A A H A A
2 ,i jn H A A
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Hebbian correlation decoding1) 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 .
2) The rare case that result in , and the correction coefficients should be discarded.
iA jA
0ijc ijc n
j iY Y , / 2i jH A A n
0ijc
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Hebbian correlation decoding3) Suppose is far away from , . In the extreme case:
, .
iA jA0ijc
ijc n ij j j ic Y nY nY
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binary vectoriA
bipolar vectoriX
sum contiguous correlation-encoded associations:
1
1 1
mT Tm i j
i
T X X X X
1 1, ,i j i jH A A H A A
1 1i i ij jj i
X T nX c X
1 1T
i i ij jj i
X T nX c X
1iX X
1iX X
Hebbian encoding method
T
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Hebbian encoding methodExample(P144): consider the three-step limit cycle:
convert bit vectors to bipolar vectors:
1 2 3 1A A A A
1 1 0 1 0A
2 1 1 0 0A
3 1 0 0 1A
1 1 1 1 1X 2 1 1 1 1X
3 1 1 1 1X
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Hebbian encoding methodProduce the asymmetric TAM matrix T:
1 2 2 3 3 1
1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1
3 1 1 11 1 1 31 3 1 11 1 3 1
T T TT X X X X X X
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Hebbian encoding methodPassing the bit vectors through T in the forward direction produces:
Produce the forward limit cycle:
iA
1 22 2 2 2 1 1 0 0AT A
2 32 2 2 2 1 0 0 1A T A
3 12 2 2 2 1 0 1 0AT A
1 2 3 1A A A A
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Competitive Learning The deterministic competitive learning law: (4-165)
(4-166) We see that the competitive learning lawuses the nonlinear forgetting term: .
ij j i ijm S S m
ij j ij i jm S m S S
j ijS m
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Competitive Learning Heb learning law uses the linear forgettingterm . 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 valueencodes the forcing signal and encodes it exponentially quickly.
1jS
ijmiS
ijm
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3.Competitive Learning.⑴ Competition as Indication;⑵ Competition as correlation detection;⑶ Asymptotic centroid estimation;⑷ Competitive covariance estimation.
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Competition as indication Centroid estimation requires that the competitive signal approximate theindicator function of the locally sampled pattern class : (4-168)
jS
jDjD
I
jj j DS y I x
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Competition as indication
If sample pattern X comes from region ,the jth competing neuron in should win, and all other competing neurons shouldLose. In practice we usually use the randomlinear competitive learning law and a simpleadditive model. (4-169)
jDYF
( ) ( )j
Tj j j DS xm f I x
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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.j jjf s
j kjf s
1
p
j kj k kk
f s S y
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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 synapticvectors, the jth competing neuron will win.
2 2
2 2
1 min
0 min
j kkj j
j kk
if x m x mS y
if x m x m
jm
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Competition as correlation detection
Using equinorm property, we can get theequivalent equalities(P147): (4-174) (4-178)
(4-179)
2 2minT Tj kk
xm xm
2 2minj kkx m x m
maxT Tj kk
xm xm
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Competition as correlation detection
From the above equality, we can get: The jthCompeting neuron wins iff the input signalor pattern correlates maximally with .The cosine law: (4-180)
x jm
cos ,Tj j jXm X m x m
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Asymptotic centroid estimationThe simpler competitive law:
(4-181)If we use the equilibrium condition: (4-182)
jj D j jm I x x m n
0 jm
jD j jI x x m n
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Asymptotic centroid estimationSolving for the equilibrium synaptic vector: (4-186)It show that equals the centroid of .
0 ( ) [ ]n jD j jRI x x m p x dx E n
( )
( )j
j
Dj
D
xp x dxm
p x dx
jm jD
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Competitive covariance estimation
Centroids provides a first-order Estimateof how the unknown probability Densityfunction behaves in the regions ,and local covariances provide a second-orderdescription.
jD( )p x
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Competitive covariance estimation
Extend the competitive learning laws toasymptotically estimate the localconditional covariance matrices : (4-187) (4-189)
denotes the centriod.
T
j j j jK E x x x x D
jK
( )j
j j jDx xp x D dx E X X D
jx jD
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Competitive covariance estimation
The fundamental theorem of estimation theory [Mendel 1987]:
(4-190) is Borel-measurable random vector function
T
T
E y E y x y E y x
E y f x y f x
( )f x
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Competitive covariance estimation
At each iteration we estimate the unknown centroidas the current synaptic vector ,In this sensebecomes an error conditional covariance matrix . the stochastic difference-equation algorithm:
(4-191-192)
jx
1j j k k jm k m k c x m k
1T
j j k k j k j jK k K k d x m k x m k K k
jmjK
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Competitive covariance estimation
denotes an appropriately decreasing sequence oflearning coefficients in(4-192). If the ith neuron loses the metricalcompetition
kd
1j jm k m k
1j jK k K k
YF
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Competitive covariance estimation
The algorithm(4-192) corresponds to thestochastic differential equation:
(4-195)
(4-199)
0 [( ) ( ) ]j
Tj D j j j jK I x x x x x K N
j
T
j D j j j jK I x m x m K N
( ) ( ) ( )
( )j
j
Tj jD
j
D
x x x x p x dxK
p x dx