random sampling from a 3-way data cube - matlab

Question

imagine a 2 x 2 x 2 three-way data cube:
data = [1 2; 3 4];
data(:,:,2) = [5 6; 7 8]
I wish to generate a row-column slice from this cube (i.e. a 2x2 matrix) in which each element of the slice is obtained by randomly sampling its 3-mode fiber (i.e. an nth mode fiber is a vector running along the nth mode/dimension/way. There are 4 3-mode fibers in this cube, one of them is f1 = [1 5], another one is f2 = [2 6] and so on). For example, one slice could turn out to be:
slice = [5 2; 3 4]
a different sampling might lead to the slice:
slice = [1 2; 7 8]
Is there a quick way to do this?
I tried using slice = datasample(data,1,3) but this function randomly picks a row-column slice from the cube (i.e. either, slice = [1 2; 3 4] or [5 6; 7 8]).

Answer 1

If I understand correctly, then it's quite simple actually. You have four 3-mode fibers and you wish to construct a 2x2 matrix where each element was sampled from the corresponding fiber.
So, you need to sample 4 times (one for each fiber) one element out of 2 (each fiber has two elements):
>> [h w fiberSize] = size(data); % make the solution more general
>> fIdx = randsample( fiberSize, h*w ); % sample with replacements
After sampling we construct the slice, for simplicity I'll "flatten" the 3D data into a 2D matrix
>> fData = reshape( data, [], fiberSize );
>> slice = fData( sub2ind( [h*w fiberSize], 1:(h*w), fIdx ) );
>> slice = reshape( slice, [h w] ); % shape the slice

Answer 2

The following solution is valid for any cube size. No toolbox required.
N = size(data,1); %//length of side of cube
r = randi(N,1,N^2)-1; %//this is the random sampling
data_permuted = permute(data,[3 1 2]); %//permute so that sampling is along first dim
slice = data_permuted((1:N:N^3)+r); %//sample using linear indexing
slice = reshape(slice.',N,N); %//reshape into a matrix

Answer 3

Try this solution for any nmode (e.g. nmode=3 is the 3-mode):
data = cat(3,[1 2; 3 4],[5 6; 7 8]);
nmode = 3;
Dn = size(data,nmode);
modeSampleInds = randi(Dn,1,numel(data)/Dn); % here is the random sample
dp = reshape(permute(data,[nmode setdiff(1:ndims(data),nmode)]), Dn, []);
outSize = size(data); outSize(nmode)=1;
slice = reshape(dp(sub2ind(size(dp),modeSampleInds,1:size(dp,2))),outSize)
Note that this does not require the Statistics Toolbox. It is also completely generalized for any matrix size, number of dimension, and "N-mode fiber".
I'd be glad to explain each line if needed.