Perceptive Kuramoto Oscillators - PeKO · PDF filePerceptive Kuramoto Oscillators - PeKO Martin Meier September 9, 2013 Martin Meier PeKO September 9, 2013 1 / 1

Perceptive Kuramoto Oscillators - PeKO

Martin Meier

September 9, 2013

Martin Meier () PeKO September 9, 2013 1 / 1

Motivation

Synchrony is a natural phenomenon


Motivation

Outline


Recap

Recap: Oscillator Network

I Previous talk: Perceptual Grouping with Oscillators

I Oscillator described by phase θ and frequency ω

I Phase update:

θ̇m = ωm +K

N

N∑n=1

Fmnsin(θn − θm)

I Frequency update:

ωm = ω0 · argmaxα

( ∑n∈N (α)

Fmn ·1

2(cos(θn − θm) + 1)

)


Recap




I Phase update:

θ̇m = ωm +K

N

N∑n=1

Fmnsin(θn − θm)

I Frequency update:


( ∑n∈N (α)

Fmn ·1

2(cos(θn − θm) + 1)

)


Recap




I Phase update:

θ̇m = ωm +K

N

N∑n=1

Fmnsin(θn − θm)

I Frequency update:


( ∑n∈N (α)

Fmn ·1

2(cos(θn − θm) + 1)

)


Recap




I Phase update:

θ̇m = ωm +K

N

N∑n=1

Fmnsin(θn − θm)

I Frequency update:


( ∑n∈N (α)

Fmn ·1

2(cos(θn − θm) + 1)

)


Recap

Recap: Evaluation

I Comparison to the CLM, similar settings for both

I IA matrix with 1000 features in ten groups

I 100 layers, 100 discrete frequencies

I All with different amounts of noise in the IA matrices

I 500 trials for each condition


Recap

Recap: Evaluation Results

I Previous talk: Evaluation results

I Evaluation revealed:

I Quality comparable to the CLMI Computational complexity reducedI Grouping speed increased


Recap


I Previous talk: Evaluation resultsI Evaluation revealed:

I Quality comparable to the CLM

I Computational complexity reducedI Grouping speed increased

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40 45

gro

up

ing q

ualit

y, m

ean a

nd

std

dev

% of noise

CLMOscillators


Recap


I Previous talk: Evaluation resultsI Evaluation revealed:

I Quality comparable to the CLMI Computational complexity reducedI Grouping speed increased

0

50

100

150

200

250

300

0 5 10 15 20 25 30# o

f up

date

ste

ps,

mean a

nd s

tddev

% of noise

CLMOscillators

31 32 33 34 35 36 37 38 39 0

100

200

300

400

500

600

700

800

900

1000

# o

f update

ste

ps,

mean a

nd s

tddev

% of noise

CLMOscillators


Recap New Evaluation

New: Robustness to Perturbations

I Both models converge for 500 steps

I Split target groups (10 → 20)

I Measure #steps needed for new grouping result


Recap New Evaluation

New: Robustness to Perturbations

I Both models converge for 500 steps

I Split target groups (10 → 20)

I Measure #steps needed for new grouping result

1

10

100

5 10 15 20 25 30 35

avera

ge n

um

ber

of

steps

% of noise

CLMOscillators


Oscillation and Order

Stablility

I PeKO achieves good grouping results.

I How to assess the grouping quality?



Kuramoto Order Parameter

I Complex parameter, r and ψ

I re iψ =1

N

N∑n=1

e iθn

I r is phase coherence ∈ [0, 1]

I ψ is average phase

Re

Im

ψ

R

complex plane



Order Parameter

I Needs to be adapted for discrete frequencies:

rαeiψα =

1

Nα

Nα∑n=1

e iθn if Nα 6= 0



Order Behavior

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40 45

gro

up

ing q

ualit

y, m

ean a

nd

std

dev

% of noise

CLMOscillators

I Recap: Grouping quality with respect to noise

I Order parameter r̄ wrt. noise and time

I Order parameter can be used to assess grouping quality



Order Behavior

0

5

10

15

20

25

30

35

40

45

0 200 400 600 800 1000

am

ount

of

nois

e

update step

0

0.2

0.4

0.6

0.8

1

ord

er

0.90.8

0.7

0.6

0.5

0.6

0.5

0.40.3






Order Behavior

0

5

10

15

20

25

30

35

40

45

0 200 400 600 800 1000

am

ount

of

nois

e

update step

0

0.2

0.4

0.6

0.8

1

ord

er

0.90.8

0.7

0.6

0.5

0.6

0.5

0.40.3





Model Extensions

Dealing with spurious features

I Everyone knows: Not every feature is relevant

I We have to deal with them

(a) IA Matrix. (b) Step 1. (c) Step 2. (d) Step 3. (e) Step 50.


Model Extensions


I Everyone knows: Not every feature is relevant

I We have to deal with them

(f) IA Matrix. (g) Step 1. (h) Step 2. (i) Step 3. (j) Step 50.


Model Extensions


I Background Layer: Nice idea “borrowed” from the CLM

I Introduced as special frequency

I Possesses “chaotic” coupling:

θ̇m = ωm + Kr sin(ψ − θm).


Model Extensions


I Same example as before

I Spurious features are collected by the background frequency

(k)Initialization.

(l) Step 1. (m) Step 2. (n) Step 3. (o) Step 50.


Model Extensions

Real World Example: Texture Grouping

(p) Input image. (q) Without background. (r) With background.


Improved Learning

Improved Learning of Lateral Interactions

I Idea from S. Weng

I Represent feature compatibility by distance functions

I Create prototypes with VQ

I Labeled examples are used to decide if +/− interaction


Improved Learning


I Original approach used Activity Equilibrium VQI Replaced by ITVQ

I Better distribution of prototypes

I Evaluated in contour grouping task


Improved Learning


I Evaluated with three kinds of shapes

I 200 trials for each shapeI Four conditions

I CLM with AEVI PeKO with AEVI CLM with ITVQI PeKO with ITVQ


Improved Learning

Feature and Data Example

(s) Oriented edge features. (t) “Easy” problem. (u) “Hard” problem.


Improved Learning

Example of generated Prototypes

C+C-

Reference

C+C-

Reference

ITVQ Prototypes AEV Prototypes


Improved Learning

Improved Learning of Lateral Interactions - Results

0

0.2

0.4

0.6

0.8

1

GroupingQuality

AEV CLMAEV Oscillators

ITVQ CLMITVQ Oscillators

50 Prototypes 100 Prototypes 150 Prototypes 200 Prototypes


Conclusion

Conclusion

I Oscillators are robust to perturbations

I Order allows assessment of grouping quality

I “Chaotic” frequency handles spurious features

I Learing of lateral interactions is improved


Conclusion

Conclusion






Conclusion

Conclusion






Conclusion

Conclusion






Conclusion

Thank You!

Any Questions?


Conclusion

Thank You!

Any Questions?


ITVQ Update Rule

I Minimize Cauchy-Schwartz Divergence

I x0 is input, x prototypes

I Fixed point update rule:

x t+1i =

∑N0j=1 Gσ(x ti − x0j)x0j∑N0j=1 Gσ(x ti − x0j)

− c

∑Nj=1 Gσ(x ti − x tj )x tj∑N0j=1 Gσ(x ti − x0j)

+c

∑Nj=1 Gσ(x ti − x tj )∑N0j=1 Gσ(x ti − x0j)

x ti ; c =N0

N

V (X ,X0)

V (X )


Learning of Lateral Interactions

+

labeled training data

1

f >0

f <0

proximityvector d(v ,v )

Voronoi cellproximity prototype

r r'

rr'

rr'

proximity space D

clustering

positive coefficents

negative coefficents interaction function

2

3


Perceptive Kuramoto Oscillators - PeKO · PDF filePerceptive Kuramoto Oscillators - PeKO Martin Meier September 9, 2013 Martin Meier PeKO September 9, 2013 1 / 1

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