CONTRACTION {PTAk}

Contraction of two tensors

Description

Computes the contraction product of two tensors as a generalisation of matrix product.

Usage

 CONTRACTION(X,z, Xwiz=NULL,zwiX=NULL,rezwiX=FALSE,usetensor=TRUE)
 CONTRACTION.list(X,zlist,moins=1,zwiX=NULL,usetensor=TRUE,withapply=FALSE)

Arguments

X

a tensor(as an array) of any order

z

another tensor (with at least one space in common)

zlist

a list of lists like a solution.PTAk at least with v for every list(here v can be any array)

Xwiz

Xwiz is to specify the entries of X to contract with entries of z specified by zwiX, if Xwiz NULL dim(z)[zwiX] matching dim(X) will do without ambiguity (taking all z dimensions if zwiX is NULL). In CONTRACTION.list it is not set as one supposes the contractions in the list to operate follow the dimensions of X

zwiX

idem as Xwiz. If both Xwiz and zwiX are NULL zwiXis replaced by full possibilities (1:length(dimz)) then Xwiz is looked for. In CONTRACTION.list it is the vector for dimensions in the v to contract with X. Only 1-way dimension for each v.

moins

the elements in zlist to skip (see also TENSELE)

rezwiX

logical if TRUE (and zwiX is NULL) rematches the dimensions in for zwiX: useful only if the dimensions of z were not following the Xwiz order and are not equals.

usetensor

if TRUE uses tensor (add-on package)

withapply

if TRUE (only for vectors in zlist uses apply

Details

Like two matrices contract according to the appropriate dimensions (columns matching rows) when one performs a matrix product, this operation does pretty much the same thing for tensors(array) and specified contraction dimensions given by Xwiz and zwiX which should match. The function is actually written like: transforms both tensors as matrices with the “matching tensor product" of their contraction dimensions in columns (for higher order tensor) and rows (the other one), performs the matrix product and rebuild the result as a tensor(array). Without using tensor, if Xwiz and/or zwiX are not specified the functions tries to match all z dimensions onto the dimensions of X where X is the higher order tensor (if it is not the case in the arguments the function swaps them).

Value

A tensor of dimension c(dim(X)[-Xwiz],dim(z)[-zwiX]) if X has got a bigger order than z.

Note

This operation generalises the matrix product to the contracted product of any two tensors(arrays), and should theoretically perform the tensor product if no matching (no contraction) but has not been implemented. I recently put the option of using tensor which does exactly the same thing faster as well as it is from C. When using tensor if Xwiz or zwiX are NULL they are replaced by the full possibilities.

Author(s)

Didier G. Leibovici

References

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.

Schwartz L (1975) Les Tenseurs. Herman, Paris.

See Also

PTAk, APSOLUk

Examples

 library(tensor)
   z <-  array(1:12,c(2,3,2))
   X <- array(1:48,c(3,4,2,2))
   Xcz <- CONTRACTION(X,z,Xwiz=c(1,3,4),zwiX=c(2,3,1))
   dim(Xcz)   # 4
   Xcz1 <- CONTRACTION(X,z,Xwiz=c(3,4),zwiX=c(1,3))
   dim(Xcz1) # 3,4,3
   Xcz2 <- CONTRACTION(X,z,Xwiz=c(3,4),zwiX=c(3,1))
   Xcz1[,,1]
   Xcz2[,,1]
   #######
   sval0 <- list(list(v=c(1,2,3,4)),list(v=rep(1,3)),list(v=c(1,3)))
   tew <- array(1:24,c(4,3,2))
    CONTRACTION.list(tew,sval0,moins=1)
       #this is equivalent to the following which may be too expensive for big datasets
    CONTRACTION(tew,TENSELE(sval0,moins=1),Xwiz=c(2,3))
   ##
     CONTRACTION.list(tew,sval0,moins=c(1,2)) #must be equal to
     CONTRACTION(tew,sval0[[3]]$v,Xwiz=3)