Evolutionary Path to Biological Kernel Machines Magnus Jändel magnus@jaendel.se Swedish Defence Research Agency.
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Evolutionary Path to Biological Kernel Machines
Magnus Jändelmagnus@jaendel.se
Swedish Defence Research Agency
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
Outline
• Support vector machine definition
• Evolutionary path to a neural SVM
• Conclusions and olfactory model
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
Support vector machine definition
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
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
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
Evolutionary Path
Stage 1
SS PR
Sensor system Simple hard-wired pattern recognizer
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
Stage 2
Sensor system Simple hard-wired pattern recognizer
SS PRx
SM
Sensory Memory
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
Stage 3
Sensor system Simple hard-wired pattern recognizer
SS PRx
SM
Sensory Memory
AM
Associative memory
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
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
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
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
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
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
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.
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
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
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
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
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
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
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
Conclusions and olfactory model
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
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
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
Magnus Jändel, Brain Inspired Cognitive Systems, 15 July 2010
Questions?
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