Vectorizing nonlinear matrix operation on many small matrices

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Adam Shaw
Adam Shaw 2020-12-18
评论： Matt J 2020-12-19
I am trying to optimize the following generic matrix operation:
m = 3; % small number in general
n = 2^20; % large power of 2 in general
A = rand(m,n);
B = zeros(m^2,m^2);
for ii = 1:size(A,2)
a = A(:,ii);
r = a*a';
B = B + kron(r,r);
end
% return B
On my computer the above takes ~7s. By compiling to a MEX file with MATLAB Coder I can improve this by ~15x. I have tried compiling to CUDA with GPU Coder, but this seems to be quite inefficient.
I think the difficulty comes from two different sources:
1) I am not sure of an efficient way to vectorize the creation of the "r" matrices from the columns of the A matrix, and so have to resort to the outer for loop approach
2) I think the Kronecker product is inefficient to implement on the gpu due to the small matrix size
The speedup from compiling to MEX is nice, but I just have this feeling that I am still doing something quite inefficiently. I would appreciate if anyone has any ideas on how to optimize the above calculation, either along the lines of the two difficulties I outlined above, or via a different approach.
  2 个评论
David Goodmanson
David Goodmanson 2020-12-19
Hi Adam,
if you replace
B = B + kron(r,r);
with
r = r(:);
BB = BB + r*r';
the loop runs about 5 times faster. (The actual substitution runs faster than that, but the nonchanged steps in the loop still of course have to be included).
Matt J
Matt J 2020-12-19
@Adam,
It may be important to know what you plan to do with B, once you've computed it.

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Matt J
Matt J 2020-12-19
编辑：Matt J 2020-12-19
Ran in:
m = 3; % small number in general
n = 2^20; % large power of 2 in general
A = rand(m,n);
tic;
B = zeros(m^2,m^2);
for ii = 1:size(A,2)
a = A(:,ii);
r = a*a';
B = B + kron(r,r);
end
toc;
Elapsed time is 6.800329 seconds.
tic;
C=reshape(A,m,1,n).*reshape(A,1,m,n);
C=reshape(C,m^2,n);
B=C*C.';
toc;
Elapsed time is 0.081757 seconds.
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Adam Shaw
Adam Shaw 2020-12-19
编辑：Adam Shaw 2020-12-19
I normally do
tic; CODE; wait(gpu); toc;
when evaluating gpu speed, which I believe is equivalent to doing gputimeit? I still am seeing about the same speedups using your gputimeit code above (using a 1660ti). I guess more specifically tcpu is ~40 ms, and tgpu is ~5 ms. Which is ~>1000x faster than the ~6 s the original method took.

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