# Optimisation algorithm RPVSCC Computes the best rank-one approximation using the RPVSCC algorithm. ```r 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-**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. ## Returns a `PTAk` object (without `datanam method`) ## References Kroonenberg P (1983) **Three-mode Principal Component Analysis: Theory and Applications**. DSWO press. Leiden. 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. ## 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 **R**echerche de la **P**remire **V**aleur **S**ingulire par **C**ontraction **C**omplte. ## Author(s) Didier G. Leibovici ## See Also `INITIA`, `PTAk`, `PCAn`, `CANDPARA`

SINGVA function