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MATLAB: How to vector-multiply two arrays of matrices?
有没有办法在 Matlab 中收缩高维张量？
例如，假设我有两个 3 维数组，大小如下:
size(A) == [M,N,P]
size(B) == [N,Q,P]
我想分别在第二个和第一个索引上收缩 A 和 B。换句话说，我想将 A 视为大小为 [M,N] 的矩阵数组，将 B 视为等长数组[N,Q] 矩阵；我想将这些数组逐个元素(逐个矩阵)相乘以获得大小为 [M,Q,P] 的东西。
我可以通过 for 循环来做到这一点:
assert(size(A,2) == size(B,1));
assert(size(A,3) == size(B,3));
M = size(A,1);
P = size(A,3);
Q = size(B,2);
C = zeros(M, Q, P);
for ii = 1:size(A,3)
C(:,:,ii) = A(:,:,ii) * B(:,:,ii);
end
有没有一种方法可以避免 for 循环？ (也许适用于任意维数的数组？)
这是一个解决方案(类似于 here 所做的)在单个矩阵乘法运算中计算结果，尽管它涉及对矩阵进行大量操作以将它们变成所需的形状。然后我将它与简单的 for 循环计算(我承认它更具可读性)进行比较
%# 3D matrices
A = rand(4,2,3);
B = rand(2,5,3);
[m n p] = size(A);
[n q p] = size(B);
%# single matrix-multiplication operation (computes more products than needed)
AA = reshape(permute(A,[2 1 3]), [n m*p])'; %'# cat(1,A(:,:,1),...,A(:,:,p))
BB = reshape(B, [n q*p]); %# cat(2,B(:,:,1),...,B(:,:,p))
CC = AA * BB;
[mp qp] = size(CC);
%# only keep "blocks" on the diagonal
yy = repmat(1:qp, [m 1]);
xx = bsxfun(@plus, repmat(1:m,[1 q])', 0:m:mp-1); %'
idx = sub2ind(size(CC), xx(:), yy(:));
CC = reshape(CC(idx), [m q p]);
%# compare against FOR-LOOP solution
C = zeros(m,q,p);
for i=1:p
C(:,:,i) = A(:,:,i) * B(:,:,i);
end
isequal(C,CC)
请注意，上面执行的乘法比需要的多，但有时 "Anyone who adds, detracts (from execution time)" .遗憾的是，情况并非如此，因为 FOR 循环在这里要快得多:)
我的观点是要表明矢量化并不容易，基于循环的解决方案并不总是不好的...