# SINGVA: Optimisation algorithm RPVSCC In PTAk: Principal Tensor Analysis on k Modes

## Description

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

## Usage

 ```1 2 3 4 5``` ```SINGVA(X,test=1E-12,PTnam="vs111",Maxiter=2000, verbose=getOption("verbose"),file=NULL, smoothing=FALSE,smoo=list(NA), modesnam=NULL, Ini="svds",sym=NULL) ```

## Arguments

 `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

## Details

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-kmodes 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-kmodes will look for best approximation according to the orthogonal rank (i.e. the rank-one tensors (of the decomposition) are orthogonal), PCA-kmodes 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-kmodes a model with nested decompositions not giving the exact orthogonal rank. So PTA-kmodes will look for best approximation according to orthogonal tensors in a nested approximmation process.

## Value

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

## Note

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 Recherche de la Premi<e8>re Valeur Singuli<e8>re par Contraction Compl<ea>te.

## Author(s)

Didier G. Leibovici

## References

Kroonenberg P (1983) Three-mode Principal Component Analysis: Theory and Applications. DSWO press. Leiden.(related references in http://three-mode.leidenuniv.nl)

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 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.

`INITIA`, `PTAk`, `PCAn`, `CANDPARA`