MATLAB优化diag（A*B）吗？_Matlab_Matrix_Linear Algebra

MATLAB优化diag（A*B）吗？

matlab matrix

MATLAB优化diag（A*B）吗？,matlab,matrix,linear-algebra,Matlab,Matrix,Linear Algebra,假设我有两个非常大的矩阵A（M-by-N）和B（N-by-M）。我需要A*B的对角线。计算完整的A*B需要M*M*N乘法，而计算它的对角线只需要M*N乘法，因为不需要计算最终会超出对角线的元素 MATLAB是否实现了这一点并自动优化了diag（A*B），或者在这种情况下使用for循环更好？是的，这是for循环更好的罕见情况之一我通过探查器运行了以下脚本： M = 5000; N = 5000; A = rand(M, N); B = rand(N, M); product = A*B; di

假设我有两个非常大的矩阵

（M-by-N）和

（N-by-M）。我需要A*B的对角线。计算完整的

A*B

需要M*M*N乘法，而计算它的对角线只需要M*N乘法，因为不需要计算最终会超出对角线的元素

MATLAB是否实现了这一点并自动优化了diag（A*B），或者在这种情况下使用for循环更好？

是的，这是for循环更好的罕见情况之一

我通过探查器运行了以下脚本：

M = 5000;
N = 5000;

A = rand(M, N); B = rand(N, M);
product = A*B;
diag1 = diag(product);

A = rand(M, N); B = rand(N, M);
diag2 = diag(A*B);

A = rand(M, N); B = rand(N, M);
diag3 = zeros(M,1);
for i=1:M
    diag3(i) = A(i,:) * B(:,i);
end

我在每次测试之间重置A和B，以防MATLAB试图通过缓存来加快速度

结果（为简洁起见编辑）：

时间呼叫线路
6.29 1 5产品=A*B；
<0.01 1 6 diag1=diag（产品）；
5.46 1 9 diag2=diag（A*B）；
1 12 diag3=零（M，1）；
1 13表示i=1:M
0.52 5000 14 diag3（i）=A（i，：）*B（：，i）；
<0.01 5000 15结束

我们可以看到，在这种情况下，for循环变量比其他两个变量快一个数量级。虽然

diag（A*B）

变型实际上比

diag（product）

变型快，但它充其量只是一个边缘

我尝试了一些不同的M和N值，在我的测试中，for循环变量只有在M=1时才会变慢。

实际上，您可以使用以下神奇功能比

for

循环更快：

对于大型矩阵（2000-by-2000和更大），这大约是我的机器上显式

for

循环的两倍，对于大于500-by-500的矩阵，速度更快

请注意，由于求和和和乘法的顺序不同，所有这些方法都会产生数值上不同的结果。

您可以只计算对角线元素而不使用循环：只需使用

sum(A.'.*B).'

或

还可以将

diag（A*B）

实现为

sum（A.*B'，2）

。让我们将其与针对这个问题建议的所有其他实现/解决方案一起进行基准测试

以下列出了作为功能实现的不同方法，用于基准测试：

和乘法-1

function out = sum_mult_method1(A,B)

out = sum(A.*B',2);

和乘法-2

function out = sum_mult_method2(A,B)

out = sum(A.'.*B).';

For-loop方法

function out = for_loop_method(A,B)

M = size(A,1);
out = zeros(M,1);
for i=1:M
    out(i) = A(i,:) * B(:,i);
end

全/直接乘法法

function out = direct_mult_method(A,B)

out = diag(A*B);

Bsxfun方法

function out = bsxfun_method(A,B)

out = sum(bsxfun(@times,A,B.'),2);

基准测试代码

num_runs = 1000;
M_arr = [100 200 500 1000];
N = 4;

%// Warm up tic/toc.
tic();
elapsed = toc();
tic();
elapsed = toc();

for k2 = 1:numel(M_arr)
    M = M_arr(k2);

    fprintf('\n')
    disp(strcat('*** Benchmarking sizes are M =',num2str(M),' and N = ',num2str(N)));

    A = randi(9,M,N);
    B = randi(9,N,M);

    disp('1. Sum-multiplication method-1');
    tic
    for k = 1:num_runs
        out1 = sum_mult_method1(A,B);
    end
    toc
    clear out1

    disp('2. Sum-multiplication method-2');
    tic
    for k = 1:num_runs
        out2 = sum_mult_method2(A,B);
    end
    toc
    clear out2

    disp('3. For-loop method');
    tic
    for k = 1:num_runs
        out3 = for_loop_method(A,B);
    end
    toc
    clear out3

    disp('4. Direct-multiplication method');
    tic
    for k = 1:num_runs
        out4 = direct_mult_method(A,B);
    end
    toc
    clear out4

    disp('5. Bsxfun method');
    tic
    for k = 1:num_runs
        out5 = bsxfun_method(A,B);
    end
    toc
    clear out5

end

结果

*** Benchmarking sizes are M =100 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.015242 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.015180 seconds.
3. For-loop method
Elapsed time is 0.192021 seconds.
4. Direct-multiplication method
Elapsed time is 0.065543 seconds.
5. Bsxfun method
Elapsed time is 0.054149 seconds.

*** Benchmarking sizes are M =200 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.009138 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.009428 seconds.
3. For-loop method
Elapsed time is 0.435735 seconds.
4. Direct-multiplication method
Elapsed time is 0.148908 seconds.
5. Bsxfun method
Elapsed time is 0.030946 seconds.

*** Benchmarking sizes are M =500 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.033287 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.026405 seconds.
3. For-loop method
Elapsed time is 0.965260 seconds.
4. Direct-multiplication method
Elapsed time is 2.832855 seconds.
5. Bsxfun method
Elapsed time is 0.034923 seconds.

*** Benchmarking sizes are M =1000 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.026068 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.032850 seconds.
3. For-loop method
Elapsed time is 1.775382 seconds.
4. Direct-multiplication method
Elapsed time is 13.764870 seconds.
5. Bsxfun method
Elapsed time is 0.044931 seconds.

中间结论

*** Benchmarking sizes are M =100 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.015242 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.015180 seconds.
3. For-loop method
Elapsed time is 0.192021 seconds.
4. Direct-multiplication method
Elapsed time is 0.065543 seconds.
5. Bsxfun method
Elapsed time is 0.054149 seconds.

*** Benchmarking sizes are M =200 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.009138 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.009428 seconds.
3. For-loop method
Elapsed time is 0.435735 seconds.
4. Direct-multiplication method
Elapsed time is 0.148908 seconds.
5. Bsxfun method
Elapsed time is 0.030946 seconds.

*** Benchmarking sizes are M =500 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.033287 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.026405 seconds.
3. For-loop method
Elapsed time is 0.965260 seconds.
4. Direct-multiplication method
Elapsed time is 2.832855 seconds.
5. Bsxfun method
Elapsed time is 0.034923 seconds.

*** Benchmarking sizes are M =1000 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.026068 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.032850 seconds.
3. For-loop method
Elapsed time is 1.775382 seconds.
4. Direct-multiplication method
Elapsed time is 13.764870 seconds.
5. Bsxfun method
Elapsed time is 0.044931 seconds.

看起来，

sum-multiply

方法是最好的方法，尽管

bsxfun

方法似乎正在追赶它们，因为

从100增加到1000

接下来，仅使用

sum multiply

和

bsxfun

方法测试更高的基准规模。尺寸是-

M_arr = [1000 2000 5000 10000 20000 50000];

结果是-

*** Benchmarking sizes are M =1000 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.030390 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.032334 seconds.
5. Bsxfun method
Elapsed time is 0.047377 seconds.

*** Benchmarking sizes are M =2000 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.040111 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.045132 seconds.
5. Bsxfun method
Elapsed time is 0.060762 seconds.

*** Benchmarking sizes are M =5000 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.099986 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.103213 seconds.
5. Bsxfun method
Elapsed time is 0.117650 seconds.

*** Benchmarking sizes are M =10000 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.375604 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.273726 seconds.
5. Bsxfun method
Elapsed time is 0.226791 seconds.

*** Benchmarking sizes are M =20000 and N =4
1. Sum-multiplication method-1
Elapsed time is 1.906839 seconds.
2. Sum-multiplication method-2
Elapsed time is 1.849166 seconds.
5. Bsxfun method
Elapsed time is 1.344905 seconds.

*** Benchmarking sizes are M =50000 and N =4
1. Sum-multiplication method-1
Elapsed time is 5.159177 seconds.
2. Sum-multiplication method-2
Elapsed time is 5.081211 seconds.
5. Bsxfun method
Elapsed time is 3.866018 seconds.

备用基准测试代码（带'timeit）

绘图

num_runs = 1000;
M_arr = [1000 2000 5000 10000 20000 50000 100000 200000 500000 1000000];
N = 4;

timeall = zeros(5,numel(M_arr));
for k2 = 1:numel(M_arr)
    M = M_arr(k2);

    A = rand(M,N);
    B = rand(N,M);

    f = @() sum_mult_method1(A,B);
    timeall(1,k2) = timeit(f);
    clear f

    f = @() sum_mult_method2(A,B);
    timeall(2,k2) = timeit(f);
    clear f

    f = @() bsxfun_method(A,B);
    timeall(5,k2) = timeit(f);
    clear f

end

figure,
hold on
plot(M_arr,timeall(1,:),'-ro')
plot(M_arr,timeall(2,:),'-ko')
plot(M_arr,timeall(5,:),'-.b')
legend('sum-method1','sum-method2','bsxfun-method')
xlabel('M ->')
ylabel('Time(sec) ->')

最终结论

*** Benchmarking sizes are M =100 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.015242 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.015180 seconds.
3. For-loop method
Elapsed time is 0.192021 seconds.
4. Direct-multiplication method
Elapsed time is 0.065543 seconds.
5. Bsxfun method
Elapsed time is 0.054149 seconds.

*** Benchmarking sizes are M =200 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.009138 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.009428 seconds.
3. For-loop method
Elapsed time is 0.435735 seconds.
4. Direct-multiplication method
Elapsed time is 0.148908 seconds.
5. Bsxfun method
Elapsed time is 0.030946 seconds.

*** Benchmarking sizes are M =500 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.033287 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.026405 seconds.
3. For-loop method
Elapsed time is 0.965260 seconds.
4. Direct-multiplication method
Elapsed time is 2.832855 seconds.
5. Bsxfun method
Elapsed time is 0.034923 seconds.

*** Benchmarking sizes are M =1000 and N =4
1. Sum-multiplication method-1
Elapsed time is 0.026068 seconds.
2. Sum-multiplication method-2
Elapsed time is 0.032850 seconds.
3. For-loop method
Elapsed time is 1.775382 seconds.
4. Direct-multiplication method
Elapsed time is 13.764870 seconds.
5. Bsxfun method
Elapsed time is 0.044931 seconds.

似乎直到某个阶段，

M=5000左右，bsxfun
之后，bsxfun
似乎才占上风
未来工作
人们可以研究不同的N
，并研究这里提到的实现的性能。
通常，这些数字有多大-N和M？我遇到这个问题的应用是，A是人工神经网络中的权重向量，B是输入向量和另一个权重矩阵之间的差。在这种特殊情况下，M将介于100和1000之间，N=4。因此，我的原始应用程序实际上可能不符合“非常大的矩阵”的条件PIf如果答案中所做的所有基准测试表明，diag（A*B）
未即时优化，请联系MathWorks并提出功能请求。+1了解此方法，以及有关数值差异的非常适当的说明。如果你有基准测试代码，也许你也可以比较我的解决方案并告诉我结果？@LuisMendo:bsxfun
方法比你的5000乘5000案例的解决方案快20%左右。我想这可能是方法中隐含的并行化。谢谢！很高兴知道。这是喜欢bsxfun
的另一个原因。但是，*
和sum
也可以从并行化中获益……或者sum（A.*B'，2）
。我有一些关于这方面的基准测试结果，即将发布！希望没问题@太棒了！我刚才问过霍奇勒。顺便说一句，我倾向于避免使用和中的，2
，因为它比较慢，但现在我不确定这种感觉是否正确。是的，我在提交答案后不久就意识到了这一点，并准备在中编辑它。：）@LuisMendo也为您的实现添加了单独的基准测试结果timeit
是我现在正在研究的，目的是验证这里给出的基准测试结果是否可靠。这些时间看起来很小，很可靠，而且你没有做任何热身。如果您不熟悉可靠的基准测试，但您最近有一个Matlab，我建议您使用timeit
。另外，测试bsxfun？@horchler这些数字是OP在评论中建议使用的。现在看看timeit。这些可能是矩阵的大小，但这并不意味着你不能做更多的运行，也许更重要的是，在计时之前预热函数。
num_runs = 1000;
M_arr = [1000 2000 5000 10000 20000 50000 100000 200000 500000 1000000];
N = 4;

timeall = zeros(5,numel(M_arr));
for k2 = 1:numel(M_arr)
    M = M_arr(k2);

    A = rand(M,N);
    B = rand(N,M);

    f = @() sum_mult_method1(A,B);
    timeall(1,k2) = timeit(f);
    clear f

    f = @() sum_mult_method2(A,B);
    timeall(2,k2) = timeit(f);
    clear f

    f = @() bsxfun_method(A,B);
    timeall(5,k2) = timeit(f);
    clear f

end

figure,
hold on
plot(M_arr,timeall(1,:),'-ro')
plot(M_arr,timeall(2,:),'-ko')
plot(M_arr,timeall(5,:),'-.b')
legend('sum-method1','sum-method2','bsxfun-method')
xlabel('M ->')
ylabel('Time(sec) ->')