Description Usage Arguments Details Value Note Author(s) References See Also
Computes the best rankone approximation using the RPVSCC algorithm.
1 2 3 4 5 
X 
a tensor (as an array) of order k, if nonidentity metrics are
used 
test 
numerical value to stop optimisation 
PTnam 
character giving the name of the kmodes Principal Tensor 
Maxiter 
if 
verbose 
control printing 
file 
output printed at the prompt if 
smoothing 
logical to use smooth functiosns or not (see

smoo 
list of functions returning smoothed vectors (see

modesnam 
character vector of the names of the modes, if 
Ini 
method used for initialisation of the algorithm (see 
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 
The algorithm termed RPVSCC in Leibovici(1993) is implemented
to compute the first Principal Tensor (rankone 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 rankone approximation issued from SVD whith
2 modes. It is identical to the PCAkmodes 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,
PTAkmodes will look for best approximation according to the
orthogonal rank (i.e. the rankone tensors (of the
decomposition) are orthogonal), PCAkmodes 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
rankone 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 PTAkmodes a model with
nested decompositions not giving the exact orthogonal rank.
So PTAkmodes will look for best approximation according to orthogonal tensors in a nested approximmation process.
a PTAk
object (without datanam method
)
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 threemodes and nmodes
PCA (Kroonenberg(1980)), i.e. defining what they called a
rank(R1,R2,...,Rn) approximation (called here space ranks,
see PCAn
). RPVSCC stands for Recherche de la Premi<e8>re
Valeur Singuli<e8>re par Contraction
Compl<ea>te.
Didier G. Leibovici
Kroonenberg P (1983) Threemode Principal Component Analysis: Theory and Applications. DSWO press. Leiden.
Leibovici D(1993) Facteurs <e0> Mesures R<e9>p<e9>t<e9>es et Analyses Factorielles : applications <e0> un suivi <e9>pid<e9>miologique. Universit<e9> de Montpellier II. PhD Thesis in Math<e9>matiques et Applications (Biostatistiques).
Leibovici D and Sabatier R (1998) A Singular Value Decomposition of a kways array for a Principal Component Analysis of multiway data, the PTAk. Linear Algebra and its Applications, 269:307329.
De Lathauwer L, De Moor B and Vandewalle J (2000) On the best rank1 and rank(R1,R2,...,Rn) approximation of higherorder tensors. SIAM J. Matrix Anal. Appl. 21,4:13241342.
Kolda T.G (2003) A Counterexample to the Possibility of an Extension of the EckartYoung LowRank Approximation Theorem for the Orthogonal Rank Tensor Decomposition. SIAM J. Matrix Analysis, 24(2):763767, Jan. 2003.
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