Decompose an N-dimensional array into outer products

The SVD gives us a way to decompose a square matrix A into a sum of outer products with a minimal number of terms. In other words, given

   [U,S,V]=svd(A);
   V=S*V;

I can reconstruct A as follows,

 A=0;
 for i=1:length(A)
   A=A+U(:,i)*V(i,:);
 end

My quesion is, is there an analog of this decomposition in N-dimensions. For a 3D array, for example, can we find a minimal decomposition into triple outer products,

 A=0;
 for i=1:?
   A=A+U(:,i)*V(i,:)*reshape(  W(:,i) 1,1,[] );
 end

Answer 1

[Christine Tobler]

Your last formula corresponds to the CP (canonical-polyadic) tensor decomposition. This is in a way the equivalent of the SVD for matrices, but its numerical properties aren't as nice: There is no direct way to compute it, and iterative methods often have a tendency to get stuck in local minima.

Another generalization of the SVD to tensors is the HOSVD (higher-order SVD), which consists of orthogonal matrices U, V, and W, and a core tensor K. This tensor is not diagonal like the matrix S in the SVD, but all the slices K(:, :, i) have decreasing Frobenius norm (same for K(:, i, :) and K(i, :, :)). This can be computed directly by applying the SVD to reshaped versions of the input tensor.

[Matt J]

Thanks, Christine. I guess there are no stock Matlab commands to compute these decompositions?

[Christine Tobler]

No stock commands in MATLAB, no. There are several MATLAB toolboxes (not written by MathWorks) available online, for example:

https://www.tensortoolbox.org/index.html

https://www.tensorlab.net/

I have used the Tensor Toolbox several years ago, but haven't tried Tensorlab myself. Both provide methods for computing the two decompositions I mentioned above.