Tensor composition

Contents

function AB = tensor_composition(A,B,P,SIZE)
        

TENSOR_COMPOSITION

This function computes the composite product of sparse tensor A with the sparse tensors contained in the cell array B in the order specified by the rows of index array P. The returned tensor AB of size SIZE is obtained by summing over the products for each row of indices.

Let the input A be an $(n+1)-$ dimensional tensor $\mathbf{A}$ and the input B be a $s-$cell array $(s\ge n)$ contain tensors $\mathbf{B}^{(1)},\dots,\mathbf{B}^{(s)}$ and the input P be an $r\times n$ index matrix $\mathbf{P}=\left[\begin{array}{c}\mathbf{p}_{1}\\\vdots\\\mathbf{p}_{r}\end{array}\right]$ . We denote $\mathbf{B}_{\mathbf{p}_j}=\mathbf{B}^{(p_{j1})}\circ\dots\circ\mathbf{B}^{(p_{jn})}$ and compute the tensor composition

$\mathbf{AB} = \sum_{j=1}^{r}\mathbf{A}\cdot\mathbf{B}_{\mathbf{p}_j},$

where $\mathbf{A} \cdot \mathbf{B} = \mathbf{A} \times_{2} \mathbf{B}^{(1)} \times_3 \mathbf{B}^{(2)}\dots \times_{n+1} \mathbf{B}^{(n)}$ is a Tucker-type product.

AB = sptensor(SIZE);
parfor j_p = 1:size(P,1)
    p = P(j_p,:);
    AB_p = tensor_product(A,B(p));
    AB = AB + AB_p;
end
AB = sptensor(AB);
        

end

function [AB] = tensor_product(A,B)
        

TENSOR_PRODUCT

This function uses the Sandia Tensor Toolbox to compute the product of sparse tensor A with the sparse tensors contained in the cell array B.

The outer product $\mathbf{B}=\mathbf{B}^{(1)}\circ\dots\circ\mathbf{B}^{(n)}$ has dimensionality $s=s_1+\dots+s_n$ . We compute the Tucker-type tensor product $\mathbf{A} \cdot\mathbf{B} = \mathbf{A} \times_{2} \mathbf{B}^{(1)} \times_3 \mathbf{B}^{(2)} \dots \times_{n+1}\mathbf{B}^{(n)}$ by multiplying the last $n$ modes of $\mathbf{A}$ with the first modes of $\mathbf{B}^{(1)},\dots,\mathbf{B}^{(n)}$ respectively. The resulting tensor has size $n+1-2n+s = s-n+1$ . The same is achieved by the following:

AB = A;
for k = 1:length(B)
    AB = sptensor(ttt(AB,B{k},2,1));
end
        

end
        

Published with MATLAB® R2023a