1. G lobal O ptimality of the S uccessive M ax B et A lgorithm USC ENITIAA de NANTES France Mohamed HANAFI and Jos M.F. TEN BERGE Department of psychology University of Groningen The Netherlands

2. G lobal O ptimality of the S uccessive M ax B et A lgorithm Summary. 1. The Successive MaxBet Problem (SMP). 2. The MaxBet Algorithm. 3. Global Optimality : Motivation/Problems. 4. Conclusions and Open questions.

3. 1. T he S uccessive M ax B et P roblem ( S.M.P ) Blocks Matrix s.p.s.d

4. order 1 1. T he S uccessive M ax B et P roblem ( S.M.P )

5. order s { 1. T he S uccessive M ax B et P roblem ( S.M.P )

6. 2. T he S uccessive M ax B et A lgorithm Ten Berge (1986,1988) Order 1 1. Take arbitrary initial unit length vectors 2. Compute : 3. rescale v k to unit length, and set u k= v k 4. Repeat steps 2 and 3 till convergence

7. Order s 2. T he S uccessive M ax B et A lgorithm Ten Berge (1986,1988) 1. Take arbitrary initial unit length vectors 2. Compute : 3. rescale v k to unit length, and set u k= v k 4. Repeat steps 2 and 3 till convergence

8. Property 1 : Convergence of the MaxBet Algorithm

9. Property 2 : Necessary Condition of Convergence

10. 3. Motivation and results 1. MaxBet Algorithm depends on the starting vector 2. MaxBet algorithm does not guarantee the computation of the global solution of SMP

11. 43 23 -13 0 -7 23 31 10 1 0 -13 10 64 -19 -2 0 1 -19 24 18 -7 0 -2 18 58 3. Motivation and results : an example

12. 42185 64023  (u)= 10621 Function value { 3846.7 5978.4  (v)= 9825.1 0.67 0.36 0.20 0.53 0.30 { Starting Vector 0.64 0.31 0.64 0.24 0.10 0.69 0.72 0.58 -0.43 -0.68 { Solution Vector 0.94 0.31 -0.92 0.35 0.11

13. 3. Motivation and results: Two Questions Q1. How can we know that the solution computed by the Maxbet algorithm is global or not ? Q2. When the solution is not global, how can we reach using this solution the global solution ?

14. 3. Motivation and Results : Proceeding Global solution of SMP Spectral properties (eigenvalues and eigenvectors) of

15. RESULT 1 Result 1

16. ELEMENTS OF PROOF (Result 1)

17. (matrix is negative semi definite) 3. Motivation and Results

18. RESULT 2 then matrix is negative semi definite Result 2

19. Suppose has a positive eigenvalue 1. w is block-normed vector 2. w is not block-normed vector 2.1. w is not block orthogonal to u 2.2. w is block orthogonal to u ELEMENTS OF PROOF (Result 2)

20. 1. w is block-normed vector w is better solution than u

21. 2. w is not block-normed vector 2.1. w is not block orthogonal to u v is better solution than u

22. w is not block-normed vector 2.2. w is block orthogonal to u

23. RESULT 2 Result 3 then matrix is negative semi definite

24. Suppose has a positive eigenvalue ELEMENTS OF PROOF (Result 3)

25. u has all elements of the same sign ELEMENTS OF PROOF (Result 3) w has all elements of the same sign

26. (matrix is negative semi definite) Result 4

27. 45 -20 5 6 16 3 -20 77 -20 -25 -8 -21 5 -20 74 47 18 -32 6 -25 47 54 7 -11 16 -8 18 7 21 -7 3 -21 -32 -11 -7 70 ELEMENTS OF PROOF (Result 4)

28. 0.49 -0.87 0.80 0.59 0.56 -0.82  (u) =378.96 Random research with 10.000.000 starting vectors  =0.48 u = ELEMENTS OF PROOF (Result 4)

29. - Possible Application in statistics : Multivariate Methods (Analysis of K sets of data ) 4. General Conclusions 1. Generalized canonical correlationAnalysis: Horst (1961) 3. Soft Modeling Approach : Estimation of latent variables under mode B Wold (1984); Hanafi (2001) 2. Rotation methods : MaxDiff, MaxBet, generalized Procrustes Analysis Gower(1975); Van de Geer(1984);Ten Berge (1986,1988)

30. - - Necessary condition for the case K=3 when matrix A has not all elements of the same sign? 4. Perspective and Little Open Question

33. Motivation: Illustration 1 MaxBet Algorithm depends on the starting vector

34. The Successive MaxBet Problem (S.M.P) and Multivariate Methods

35. Some multivarite methods Generalized canonical correlation methods Rotation methods(Agreement methods) SOFT MODELING APPRAOCH(Approch)

36. Rotation methods S M P = MaxBet method Van de Geer (1984) Ten Berge (1986,1988) S M P = MaxBet method Van de Geer (1984) Ten Berge (1986,1988) S M P = MaxDiff method Van de Geer (1984) Ten Berge (1986,1988) S M P = MaxBet method Van de Geer (1984) Ten Berge (1986,1988) S M P = Generalized Procrustes Analysis Gower(1975), Ten Berge (1986,1988)

37. Generalized canonical correlation methods SVD SMP = Horst method(1961) S M P = Soft Modeling Appraoch (Hanafi 2001) Mode B soft modeling approach

40. Multivariate Eigenvalue Problem Watterson and Chu(1993)