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Martin08
Martin08

Reputation: 21460

How to vectorize this matrix multiplication in matlab

I have 2 matrices A (nxm) and B (nxd) and want to multiply element-wise each column of A with a row of B. There are m columns in A and n 1xd vectors in B so the results are m nxd matrices. Then I want to sum(result_i, 1) to get m 1xd vectors, which I want to apply vertcat to get a mxd matrix. I'm doing this operations using for loop and it is slow because n and d are big. How can I vectorize this in matlab to make it faster? Thank you.

EDIT:
You're all right: I was confused by my own question. What I meant by "multiply element-wise each column of A with a row of B" is to multiply n elements of a column in A with the corresponding n rows of B. What I want to do with one column of A is as followed (and I repeat this for m columns of A, then vertcat the C's vector together to get an mxd matrix):

column_of_A =

     3
     3
     1


B =

     3     1     3     3
     2     2     1     2
     1     3     3     3


C = sum(diag(column_of_A)*B, 1)

     16    12    15    18

Upvotes: 2

Views: 5103

Answers (3)

antman
antman

Reputation: 1

This is the way I would do this:

sum(repmat(A,1,4).*B)

If you don't know the number of columns of B:

sum(repmat(A,1,size(B,2)).*B)

Upvotes: 0

Jonas
Jonas

Reputation: 74930

You can vectorize your operation the following way. Note, however, that vectorizing comes at the cost of higher memory usage, so the solution may end up not working for you.

%# multiply nxm A with nx1xd B to create a nxmxd array
tmp = bsxfun(@times,A,permute(B,[1 3 2]));

%# sum and turn into mxd
out = squeeze(sum(tmp,1));

You may want to do everything in one line, which may help the Matlab JIT compiler to save on memory.

EDIT

Here's a way to replace the first line if you don't have bsxfun

[n,m] = size(A);
[n,d] = size(B);
tmp = repmat(A,[1 1 d]) .* repmat(permute(B,[1 3 2]),[1,m,1]);

Upvotes: 6

AVH
AVH

Reputation: 11507

It's ugly, but as far as I can see, it works. I'm not sure it will be faster than your loop though, plus, it has a large memory overhead. Anyway, here goes:

A_3D = repmat(reshape(A, size(A, 1), 1, size(A, 2)), 1, size(B, 2));
B_3D = repmat(B, [ 1 1 size(A, 2)]);
result_3D = sum(A_3D .* B_3D, 1);
result = reshape(result_3D, size(A, 2), size(B, 2))

What it does is: make A into a 3D matrix of size n x 1 x m, so one column in each index of the 3rd dimension. Then we repeat the matrix so we get an n x d x m matrix. We repeat B in the 3rd dimension as well. We then do a piecewise multiplication of all the elements and sum them. The resulting matrix is a 1 x d x m matrix. We reshape this into a m x d matrix.

I'm pretty sure I switched around the size of the dimensions a few times in my explanation, but I hope you get the general gist.

Multiplying with a diagonal matrix seems at least twice as fast, but I couldn't find a way to use diag, since it wants a vector or 2D matrix as input. I might try again later tonight, I feel there must be a faster way :).

[Edit] Split up the command in parts to at least make it a little bit readable.

Upvotes: 1

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