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2. The Generalised Mapping Regressor (GMR) neural network for inverse discontinuous problems Student : Chuan LU Promotor : Prof. Sabine Van Huffel Daily Supervisor : Dr. Giansalvo Cirrincione

3. Mapping Approximation Problem • Feedforward neural networks are : • universal approximators of nonlinear continuousfunctions (many-to-one, one-to-one) • they don’t yield multiple solutions • they don’t yield infinite solutions • they don’t approximate mapping discontinuities

4. conditional average of the target data Inverse and Discontinuous Problems • Mapping : multi-valued, complex structure. • Poor representation of the mapping by least squares approach (sum-of-squares error function) for feedforward neural networks. • Mapping with discontinuities.

5. kernel blending winner-take-all • Jacobs and Jordan • Bishop (ME extension) output mixture-of-experts gating network Network 1 Network 2 Network 3 input It partitions the solution between several networks. It uses a separate network to determine the parameters of each kernel, with a further network to determine the coefficients.

6. ME MLP Example #1

7. ME MLP Example #2

8. ME MLP Example #3

9. ME MLP Example #4

10. Characteristics: • approximate every kind of function or relation. • input : collection of components of x and youtput : estimation of the remaining components • output all solutions, mapping branches, equilevel hypersurfaces. Generalised Mapping Regressor( GMR ) (G. Cirrincione and M. Cirrincione, 1998)

11. clusters mapping branches GMR Basic Ideas function approximation pattern recognition Z (augmented) space  unsupervised learning • coarse-to-fine learning • incremental • competitive • based on mapping recovery (curse of dimensionality) • topological neuron linking • distance • direction • linking tracking • branches • contours • open architecture

12. Training Set Learning Object Merging Recall-ing Linking object 1 branch 1 branch 2 links object merged INPUT pool of neurons object 3 object 2 GMR four phases

13. vigilance threshold x w4= x4 EXIN Segmentation Neural Network (EXIN SNN) (G. Cirrincione, 1998) • clustering Input/weight space

14. branch (object) neuron GMR Learning • EXIN SNN • high rz ( say r1 ) coarse quantization Z (augmented) space

15. GMR Learning • production phase • Voronoi sets domain setting Z (augmented) space

16. GMR Learning TS#3 TS#5 TS#1 TS#4 TS#2 • secondary EXIN SNNs • rz = r2 < r1 fine quantization Z (augmented) space Other levels are possible

17. GMR Coarse to fine Learning ( Example) object neuron Voronoi set fine VQ neurons object neuron

18. asymmetric radius ri neuron i Task 1 : GMR Linking • Voronoi set: setup of the neuron radius (domain variable)

19. Task 2 : k-nn branch and bound search technique Weight Space Linking candidates w3 w4 w5 d3 Linking direction d4 d5 w1 d1 d1 d2 w2 GMR Linking • distance test • direction test • create a link or strengthen a link • For one TS presentation: zi

20. Branch and Bound Accelerated Linking • neuron tree constructed during learning phase (multilevel EXIN SNN learning) • methods in linking candidate step (k-nearest-neighbors computation): • -BnB : <  d1 , ( : linking factorpredefined) • k-BnB : k predefined.

21. 83 % GMR Linking branch-and-boundin linkingexperimental results:

22. branch and bound (cont.) Apply branch and bound in learning phase ( labelling ) : • Tree construction • k-means • EXIN SNN • Experimental results (in the 3-D example) • 50% of labeling flops are saved

23. GMR Linking Example link

24. GMR Merging Example

25. level 1 neuron branch 1 level 2 neuron branch 2 GMR Recalling Example • level one neurons : input within theirdomain • level two neurons : only connected ones • level zero neurons : isolated (noise)

26. Experiments spiral of Archimedes  = a (a = 1)

27. Sparse regions further normalizing + higher mapping resolution Experiments

28. noisy data Experiments

29. Experiments

30. GMR mapping of 8 spheres in a 3-D scene. Experiments contours: links among level one neurons

31. Conclusions GMR is able to : • solve inversediscontinuous problems • approximate every kind of mapping • yield all the solutions and the corresponding branches GMR can be accelerated by applying tree search techniques GMR needs: • interpolation techniques • kernels or projection techniques for high dimensional data • adaptive parameters

32. Thank you ! (shi-a shi-a)

33. w8 w7 l8= 0 b8 = 0 l7 = 0 b7 = 0 l1 = 0 b1 = 0 l3 = 0 b3 = 0 w3 l4 = 0 b4 = 0 w1 input w2 w4 l2= 0 b2= 0 w6 l5 = 0 b5 = 0 w5 l6 = 0 b6 = 0 connected neuron : level zero level two branch  the winner branch GMR Recall l1 = 1 b1 = 1 • restricted distance r1 l3 = 2 b3 = 1 • level one test • linking tracking

34. input w8 • level one test • linking tracking GMR Recall w7 l8= 0 b8 = 0 l7 = 0 b7 = 0 l1 = 1 b1 = 1 l1 = 0 b1 = 0 l3 = 2 b3 = 1 l3 = 0 b3 = 0 w3 l4 = 0 b4 = 0 w1 w2 r2 w4 branch cross l2= 1 b2= 2 l2= 0 b2= 0 l2= 1 b2=1 w6 l5 = 0 b5 = 0 w5 l6 = 0 b6 = 0

35. input Two Branches Tow Branches w8 GMR Recall w7 l8= 0 b8 = 0 … until completion of the candidates l7 = 0 b7 = 0 l1 = 0 b1 = 0 l1 = 1 b1 = 1 l3 = 0 b3 = 0 l3 = 2 b3 = 1 w3 l4 = 0 b4 = 0 l4 = 1 b4 = 4 w1 w2 w4 l2= 0 b2= 0 l2= 1 b2= 1 l2= 1 b2= 2 w6 l4 = 1 b4 = 5 l5 = 2 b5 = 4 l5 = 0 b5 = 0 l4 = 1 b4 = 4 w5 l6 = 2 b6 = 4 l6 = 1 b6 = 6 l6 = 0 b6 = 0 l6 = 1 b6 = 4 • level one neurons : input within theirdomain • level two neurons : only connected ones • level zero neurons : isolated (noise) clipping

36. l1 = 1 b1 = 1 l3 = 2 b3 = 1 l4 = 1 b4 = 4 input l2= 1 b2= 1 l4 = 1 b4 = 4 l6 = 1 b6 = 4 w8 • Output= weight complements of the level one neurons GMR Recall w7 l8= 0 b8 = 0 • Outputinterpolation l7 = 0 b7 = 0 w3 w1 w2 w4 w6 w5