[英]Is it possible to speed up this MATLAB script?
I've encountered some performance problems thus I want to speed up those running-slow scripts. 我遇到了一些性能问题,因此我想加快那些运行缓慢的脚本。 But I have no more ideas on how to speed up them. 但我对如何加快它们没有更多的想法。 Because I found I was often blocked with the indices. 因为我发现我经常被指数所阻挡。 I found the abstract thinking is very difficult for me. 我发现抽象思维对我来说非常困难。
The script is 脚本是
tic,
n = 1000;
d = 500;
X = rand(n, d);
R = rand(n, n);
F = zeros(d, d);
for i=1:n
for j=1:n
F = F + R(i,j)* ((X(i,:)-X(j,:))' * (X(i,:)-X(j,:)));
end
end
toc
Few approaches with bsxfun
could be suggested here. 这里可以建议使用bsxfun
方法很少。 Also, read on to see how one can get 30x+
speedup on a problem like this! 另外,继续阅读以了解如何在这样的问题上获得30x+
加速!
Approach #1 (Naive vectorized approach) 方法#1(天真矢量化方法)
To accommodate the two operations of subtractions between rows of X
and then the subsequent element-wise multiplications between them, a naive bsxfun
based approach would lead to a 4D intermediate array which would correspond to ((X(i,:)-X(j,:))' * (X(i,:)-X(j,:)))
. 为了适应X
行之间的两次减法操作,然后是它们之间的后续逐元素乘法,一个朴素的基于bsxfun
的方法将导致一个4D中间数组,它对应于((X(i,:)-X(j,:))' * (X(i,:)-X(j,:)))
bsxfun
((X(i,:)-X(j,:))' * (X(i,:)-X(j,:)))
。 After that, one needs to multiply R
to have the final output F
. 之后,需要将R
乘以得到最终输出F
This is implemented as shown next - 这是如下所示实现的 -
v1 = bsxfun(@minus,X,permute(X,[3 2 1]));
v2 = bsxfun(@times,permute(v1,[1 3 2]),permute(v1,[1 3 4 2]));
F = reshape(R(:).'*reshape(v2,[],d^2),d,[]);
Approach #2 (Not-so-naive vectorized approach) 方法#2(不那么天真的矢量化方法)
The earlier mentioned approach goes into 4D which could slow down things. 前面提到的方法进入4D可能会减慢速度。 So, instead you can keep the intermediate data until 3D by reshaping. 因此,您可以通过重新整形将中间数据保留到3D。 This is listed next - 这是下一个 -
sub1 = bsxfun(@minus,X,permute(X,[3 2 1]));
sub1_2d = reshape(permute(sub1,[1 3 2]),n^2,[])
mult1 = bsxfun(@times,sub1_2d,permute(sub1_2d,[1 3 2]))
F = reshape(R(:).'*reshape(mult1,[],d^2),d,[])
Approach #3 (Hybrid approach) 方法#3(混合方法)
Now, you can make a hybrid approach based on Approach #2 ( vectorized subtractions
+ loopy multiplications
). 现在,您可以基于方法#2 ( vectorized subtractions
+ loopy multiplications
)制作混合方法。 Benefit of this approach would be that it uses the fast matrix multiplication
to perform the multiplications and reduces the complexity to O(n) from the earlier O(n^2) and this should make it much more efficient. 这种方法的好处在于它使用fast matrix multiplication
来执行乘法,并将复杂度从较早的O(n ^ 2)降低到O(n),这应该使其更有效。 Thanks to @Dev-iL, for suggesting this idea! 感谢@ Dev-iL,提出这个想法! Here's the code - 这是代码 -
sub1 = bsxfun(@minus,X,permute(X,[3 2 1]));
sub1 = bsxfun(@times,sub1,permute(sqrt(R),[1 3 2]));
F = zeros(d);
for k = 1:size(sub1,3)
blk = sub1(:,:,k);
F = F + blk.'*blk;
end
Benchmarking code comparing the original approach against Approach #3 比较原始方法与方法#3的基准代码
%// Parameters
n = 500;
d = 250;
X = rand(n, d);
R = rand(n, n);
%// Warm up tic/toc.
for k = 1:100000
tic(); elapsed = toc();
end
disp('------------------------------ With Original Approach')
tic
F1 = zeros(d, d);
for i=1:n
for j=1:n
F1 = F1 + R(i,j)*((X(i,:)-X(j,:))' * (X(i,:)-X(j,:)));
end
end
toc, clear F1 i j
disp('------------------------------ With Proposed Approach #3')
tic
sub1 = bsxfun(@minus,X,permute(X,[3 2 1]));
sub1 = bsxfun(@times,sub1,permute(sqrt(R),[1 3 2]));
F = zeros(d);
for k = 1:size(sub1,3)
blk = sub1(:,:,k);
F = F + blk.'*blk;
end
toc
Runtime results 运行时结果
------------------------------ With Original Approach
Elapsed time is 29.728571 seconds.
------------------------------ With Proposed Approach #3
Elapsed time is 0.839726 seconds.
So, who's ready for a 30x+ speedup!? 那么,谁准备好了30倍以上的加速!?
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