Multiply a 3D matrix with a 2D matrix

You can do this in one line using the functions NUM2CELL to break the matrix X into a cell array and CELLFUN to operate across the cells:

Z = cellfun(@(x) x*Y,num2cell(X,[1 2]),'UniformOutput',false);

The result Z is a 1-by-C cell array where each cell contains an A-by-D matrix. If you want Z to be an A-by-D-by-C matrix, you can use the CAT function:

Z = cat(3,Z{:});

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NOTE: My old solution used MAT2CELL instead of NUM2CELL, which wasn't as succinct:

[A,B,C] = size(X);
Z = cellfun(@(x) x*Y,mat2cell(X,A,B,ones(1,C)),'UniformOutput',false);

I'm approaching the exact same issue, with an eye for the most efficient method. There are roughly three approaches that i see around, short of using outside libraries (i.e., mtimesx):

1. Loop through slices of the 3D matrix
2. repmat-and-permute wizardry
3. cellfun multiplication

I recently compared all three methods to see which was quickest. My intuition was that (2) would be the winner. Here's the code:

% generate data
A = 20;
B = 30;
C = 40;
D = 50;

X = rand(A,B,C);
Y = rand(B,D);

% ------ Approach 1: Loop (via @Zaid)
tic
Z1 = zeros(A,D,C);
for m = 1:C
    Z1(:,:,m) = X(:,:,m)*Y;
end
toc

% ------ Approach 2: Reshape+Permute (via @Amro)
tic
Z2 = reshape(reshape(permute(X, [2 1 3]), [A B*C]), [B A*C])' * Y;
Z2 = permute(reshape(Z2',[D A C]),[2 1 3]);
toc


% ------ Approach 3: cellfun (via @gnovice)
tic
Z3 = cellfun(@(x) x*Y,num2cell(X,[1 2]),'UniformOutput',false);
Z3 = cat(3,Z3{:});
toc

All three approaches produced the same output (phew!), but, surprisingly, the loop was the fastest:

Elapsed time is 0.000418 seconds.
Elapsed time is 0.000887 seconds.
Elapsed time is 0.001841 seconds.

Note that the times can vary quite a lot from one trial to another, and sometimes (2) comes out the slowest. These differences become more dramatic with larger data. But with much bigger data, (3) beats (2). The loop method is still best.

% pretty big data...
A = 200;
B = 300;
C = 400;
D = 500;
Elapsed time is 0.373831 seconds.
Elapsed time is 0.638041 seconds.
Elapsed time is 0.724581 seconds.

% even bigger....
A = 200;
B = 200;
C = 400;
D = 5000;
Elapsed time is 4.314076 seconds.
Elapsed time is 11.553289 seconds.
Elapsed time is 5.233725 seconds.

But the loop method can be slower than (2), if the looped dimension is much larger than the others.

A = 2;
B = 3;
C = 400000;
D = 5;
Elapsed time is 0.780933 seconds.
Elapsed time is 0.073189 seconds.
Elapsed time is 2.590697 seconds.

So (2) wins by a big factor, in this (maybe extreme) case. There may not be an approach that is optimal in all cases, but the loop is still pretty good, and best in many cases. It is also best in terms of readability. Loop away!

As a personal preference, I like my code to be as succinct and readable as possible.

Here's what I would have done, though it doesn't meet your 'no-loops' requirement:

for m = 1:C

    Z(:,:,m) = X(:,:,m)*Y;

end

This results in an A x D x C matrix Z.

And of course, you can always pre-allocate Z to speed things up by using Z = zeros(A,D,C);.

Here's a one-line solution (two if you want to split into 3rd dimension):

A = 2;
B = 3;
C = 4;
D = 5;

X = rand(A,B,C);
Y = rand(B,D);

%# calculate result in one big matrix
Z = reshape(reshape(permute(X, [2 1 3]), [A B*C]), [B A*C])' * Y;

%'# split into third dimension
Z = permute(reshape(Z',[D A C]),[2 1 3]);

Hence now: Z(:,:,i) contains the result of X(:,:,i) * Y

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Explanation:

The above may look confusing, but the idea is simple. First I start by take the third dimension of X and do a vertical concatenation along the first dim:

XX = cat(1, X(:,:,1), X(:,:,2), ..., X(:,:,C))

... the difficulty was that C is a variable, hence you can't generalize that expression using cat or vertcat. Next we multiply this by Y:

ZZ = XX * Y;

Finally I split it back into the third dimension:

Z(:,:,1) = ZZ(1:2, :);
Z(:,:,2) = ZZ(3:4, :);
Z(:,:,3) = ZZ(5:6, :);
Z(:,:,4) = ZZ(7:8, :);

So you can see it only requires one matrix multiplication, but you have to reshape the matrix before and after.