evolutionary path to biological kernel machines magnus jändel [email protected] swedish defence...
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Evolutionary Path to Biological Kernel Machines
Magnus Jä[email protected]
Swedish Defence Research Agency
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Summary
• It is comparatively easy for organisms to implement support vector machines.
• Biological support vector machines provide efficient and cost-effective pattern recognition with one-shot learning [1].
• The support vector machine hypothesis is consistent with the architecture of the olfactory system [1].
• Bursts in the thalamocortical system may be related to support vector machine pattern recognition [2].
• An efficient implementation reuses machinery for learning action sequences [3].
1) Jändel, M.: A neural support vector machine. Neural Networks 23, 607-613 (2010).2) Jändel, M.: Thalamic bursts mediate pattern recognition. Proceedings of the 4th International IEEE EMBS Conference on Neural Engineering 562–565 (2009).3) Jändel, M.: Pattern recognition as an internalized motor programme. To appear in proc. of ICNN 2010.
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
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Outline
• Support vector machine definition
• Evolutionary path to a neural SVM
• Conclusions and olfactory model
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
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Support vector machine definition
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Maximum margin linear classification
1
( )m
i i ii
f b y b
x w x x x
Consider binary classification with m training examples: ( , ),i iyx {1, 1}iy
( ) 0f x
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
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Transform to high-dimensional feature space
1 1
( ) ( ) ( ) ( ) ( , )m m
i i i i i ii i
f b y b y K b
x w φ x φ x φ x x x
1
( ) ( , )m
i i ii
f y K
x x xZero-bias SVM:
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
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Zero-bias -SVM
, 1
1( ) ( , )
2
m
i j i j i j
i j
W y y K
α x x
10 i m
1
m
ii
Maximize:
Subject to: and
0 1 where
Solve by iterative gradient ascent in the -space hyperplane
1
2
1
1,
m
i s is
C Cm
1
( , ).m
i i j j i jj
C y y K
x xwhere The margin of the i:th example in feature space!
1
( ) ( , )m
i i ii
f y K
x x xClassification function:
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
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Evolutionary Path
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Stage 1
SS PR
Sensor system Simple hard-wired pattern recognizer
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Stage 2
Sensor system Simple hard-wired pattern recognizer
SS PRx
SM
Sensory Memory
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Stage 3
Sensor system Simple hard-wired pattern recognizer
SS PRx
SM
Sensory Memory
AM
Associative memory
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Stage 4
SS PRx
SM
AM
x
y´- Significant patterns and the associated valence are stored in the AM.- Sufficiently similar inputs make the AM recall the valence of a stored pattern.
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
Zero-bias -SVM
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Stage 5
SS PRx
SM
AM
xx´, y´
- Significant patterns and the associated valence are stored in the AM.- Sufficiently similar inputs make the AM recall the valence of a stored pattern - The PR modulates the recalled valence y´ with a similarity measure comparing input x with the storedpattern x´ according to,
( ) ( , )f y K x x x
( , )y K x x
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Stage 6
1
( ) ( , )m
i i ii
f y K
x x x
SS PRx
SM
OM
xxi, yi
- The OM oscillates between memory states - The PR computes a weighted average over the valences of all stored examples,
( )f x
0
0( ) ( )( ) ( , )
trapt T
i t i tt
f y K dt
x x x
Stage 6 implements the classification function of a zero-bias SVM.Zero-bias -SVM
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Oscillating Associative Memory
Hopfield associative memory
N neurons with binary output zi
Update rule 1
sgn( )N
i ij jj
z w z
Imprint m memory patterns x(k)
( ) ( )
1
1mk k
ij i jk
w x xN
One-shot learning!
OM Model
m memory patterns
The probability of finding the OM in state i is,
Each oscillation selects the next state with uniform probability.
The average endurance time of state i is Ti.
1
/m
i i i ii
p T T
Oscillating memory
- Firing cell nuclei are exhausted
- Active synapses are depleted
Modes with perpetual oscillation between attractors.
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Stage 7
( , )ij j i jB y K x x
SS PRxj
SM
OM
xjxi, yi- Learning feedback Bij tunes memory weights
- Real-world experiments are required
( )f x
Bij
xi is the present example presented by the OMxj is the sensory inputyj is the valence of xj as learnt from hard-earned experience
feedback
For each OM oscillation apply the learning rules,
i i ijT y B and1
: s i ijs T y Bm
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Stage 8
SS PRxj
SM
OM
xj xi, yi
- OM patterns are set up in sensory memory while sleeping- OM weights tuned in virtual experiments- No need for external feedback- Implements a zero-bias -SVM
( )f x
Bijxi
Zero-bias -SVM
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Learning SVM weights
( , )ij j i jB y K x x
1
1,
m
i s is
C Cm
For each OM oscillation apply the learning rules,
i i ijy B and1
: s i ijs y Bm
where
Averaging over “trapped examples” with probability distribution j jp
SS PRxj
SM
OM
xj xi, yiBijxi
gives
1
( , ).m
i i j j i jj
C y y K
x xwhere
Zero-bias -SVM
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Conclusions and olfactory model
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Summary of support vector machine implementation
Classification process
SS PRxj
SM
OM
xj xi, yi
( )f x
Bijxi
x
Learning new training examples
Learning weights of training examples
Zero-bias -SVM
Research program
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TrapCL
OM
OB
AOCAPC
HOBS
PPC
D1
M2
D3
D5
M1
D2
M3
D4
Olfactory model
SS PRxj
SM
OM
xj xi, yi
( )f x
Bijxi
x
APC – Anterior piriform cortexPPC – Posterior piriform cortexAOC – Anterior olfactory cortexOB – Olfactory bulbHOBS – Higher-order brain systems
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Questions?