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

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.

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

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

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

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

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

Property 1 : Convergence of the MaxBet Algorithm

Property 2 : Necessary Condition of Convergence

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

Motivation and results : an example

 (u)= Function value {  (v)= { Starting Vector { Solution Vector

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 ?

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

RESULT 1 Result 1

ELEMENTS OF PROOF (Result 1)

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

RESULT 2 then matrix is negative semi definite Result 2

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)

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

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

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

RESULT 2 Result 3 then matrix is negative semi definite

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

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

(matrix is negative semi definite) Result 4

ELEMENTS OF PROOF (Result 4)

 (u) = Random research with starting vectors  =0.48 u = ELEMENTS OF PROOF (Result 4)

- 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)

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

Motivation: Illustration 1 MaxBet Algorithm depends on the starting vector

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

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

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)

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

Multivariate Eigenvalue Problem Watterson and Chu(1993)