SINGVA                 package:PTAk                 R Documentation

_O_p_t_i_m_i_s_a_t_i_o_n _a_l_g_o_r_i_t_h_m _R_P_V_S_C_C

_D_e_s_c_r_i_p_t_i_o_n:

     Computes the best rank-one approximation using the RPVSCC
     algorithm.

_U_s_a_g_e:

     SINGVA(X,test=1E-12,PTnam="vs111",Maxiter=2000,
                       verbose=getOption("verbose"),file=NULL,
                         smoothing=FALSE,smoo=list(NA),
                          modesnam=NULL,
                           Ini="Presvd",sym=NULL)

_A_r_g_u_m_e_n_t_s:

       X: a tensor (as an array) of order _k_, if non-identity metrics
          are used 'X' is a list with 'data'  as the array and 'met' a
          list of metrics

    test: numerical value to stop optimisation

   PTnam: character giving the name of the _k_-modes Principal Tensor

 Maxiter: if 'iter > Maxiter' prompts to carry on or not, then do it
          every other 200 iterations

 verbose: control printing

    file: output printed at the prompt if 'NULL', or printed in the
          given  'file'

smoothing: logical to use smooth functiosns or not (see 'SVDgen') 

    smoo: list of functions returning smoothed vectors (see 'PTA3') 

modesnam: character vector of the names of the modes, if 'NULL' "'mo
          1'" ..."'mo k'"

     Ini: method used for initialisation of the algorithm (see
          'INITIA')

     sym: description of the symmetry of the tensor _e.g._ c(1,1,3,4,1)
          means the second mode and the fifth are identical to the
          first 

_D_e_t_a_i_l_s:

     The algorithm termed _RPVSCC_  in Leibovici(1993) is implemented
     to compute the first Principal Tensor (rank-one tensor with its
     singular value) of the given tensor 'X'. According to the
     decomposition described in Leibovici(1993) and Leibovici and
     Sabatier(1998), the function gives a generalisation  to _k_ modes
     of the _best rank-one approximation_ issued from SVD whith 2
     modes. It is identical to  the PCA-_k_modes if only 1 dimension is
     asked in each space, and to PARAFAC/CANDECOMP if the rank of the
     approximation is fixed to 1. Then the methods differs,
     PTA-_k_modes will look for best approximation according to the
     _orthogonal rank_ (_i.e._ the rank-one tensors (of the
     decomposition) are orthogonal), PCA-_k_modes will look for best
     approximation according to the _space ranks_ (_i.e._  ranks of
     every bilinear form deducted from the original tensor, that is the
     number of components in each space), PARAFAC/CANDECOMP will look
     for best approximation according to the _rank_ (_i.e._ the
     rank-one tensors are not necessarily orthogonal). 
      Recent work from Tamara G Kolda showed on an example that
     _orthogonal rank_ decompositions are not necesseraly nested. This
     makes PTA-_k_modes a model with nested decompositions not giving
     the exact _orthogonal rank_. So PTA-_k_modes will look for best
     approximation according to orthogonal tensors in a nested
     approximmation process.

_V_a_l_u_e:

     a 'PTAk' object (without 'datanam method')

_N_o_t_e:

     The algorithm was derived in generalising the _transition
     formulae_ of SVD (Leibovici 1993), can also be understood as a
     generalisation of the _power method_ (De Lathauwer et al. 2000).
     In this paper they also use a similar algorithm  to build bases in
     each space, reminiscent of three-modes and _n_-modes PCA
     (Kroonenberg(1980)), _i.e._ defining what they called a
     rank-(R1,R2,...,Rn) approximation (called here _space ranks_, see
     'PCAn'). _RPVSCC_ stands for  _R_echerche de la _P_remire
     _V_aleur _S_ingulire par _C_ontraction _C_omplte.

_A_u_t_h_o_r(_s):

     Didier Leibovici c3s2i@free.fr

_R_e_f_e_r_e_n_c_e_s:

     Kroonenberg P (1983) _Three-mode Principal Component Analysis:
     Theory and Applications_. DSWO press. Leiden.(related references
     in <URL: http://www.fsw.leidenuniv.nl/~kroonenb/>) \ Leibovici D
     (1993) _Facteurs  Mesures Rptes et Analyses Factorielles :
     applications  un suivi pidmiologique_. Universit de
     Montpellier II. PhD Thesis in Mathmatiques et Applications
     (Biostatistiques).

     Leibovici D and Sabatier R (1998) _A Singular Value Decomposition
     of a k-ways array for a Principal Component Analysis of multi-way
     data, the PTA-k_. Linear Algebra and its Applications,
     269:307-329.

     De Lathauwer L, De Moor B and Vandewalle J (2000) _On the best
     rank-1 and rank-(R1,R2,...,Rn) approximation of higher-order
     tensors_. SIAM J. Matrix Anal. Appl. 21,4:1324-1342.

     Kolda T.G (2003)_ A Counterexample to the Possibility of an
     Extension of the Eckart-Young Low-Rank Approximation Theorem for
     the Orthogonal Rank Tensor Decomposition_. SIAM J. Matrix
     Analysis, 24(2):763-767, Jan. 2003.

_S_e_e _A_l_s_o:

     'INITIA', 'PTAk', 'PCAn', 'CANDPARA'

