Evolutionary Path to Biological Kernel Machines Magnus Jändel [email protected] Swedish Defence Research Agency

Evolutionary Path to Biological Kernel Machines

Magnus Jä[email protected]

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


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


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:


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:


Evolutionary Path

Stage 1

SS PR

Sensor system Simple hard-wired pattern recognizer


Stage 2


SS PRx

SM

Sensory Memory


Stage 3


SS PRx

SM

Sensory Memory

AM

Associative memory


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.


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


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


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.


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


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


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


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


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


Questions?

Documents

Evolutionary Path to Biological Kernel Machines Magnus Jändel [email protected] Swedish Defence Research Agency