两个张量之间的克罗内克积

Question

我有两个张量：x 是 2×2×3，y 也是 2×2×3。定义张量的每个正面切片为 x1 x2 x3,y1,y2,y3。 xi 或 yi 是 2×2 矩阵。我怎样才能在 matlab 中做 x 和 y 之间的克罗内克积？我想得到的是 kron(x1,y1),kron(x2,y2),kron(x3,y3) 同时在 matlab 中没有任何循环。

Answer 1

这适用于 x 和 y 的任意大小：

构建一个 5 维数组 z。前两个维度包含 x 和 y 行组合的乘积；接下来的两个包含 x 和 y 列组合的乘积；第五个是原来的三维
该数组被重塑，一方面折叠第一个和第二个维度，另一方面折叠第三个和第四个维度，以产生最终结果：

代码：

z = bsxfun(@times, permute(x, [4 1 5 2 3]), permute(y, [1 4 2 5 3]));  %// step 1
z = reshape(z, size(x,1)^2, size(x,2)^2, size(x,3));                   %//step 2

Answer 2

这可能是一种方法 -

%// Pre-processing part
[m,n,r] = size(x) %// Get size
N = m*n %// number of elements in one 3D slice

%// ------------- PART 1: Get indices for each 3D slice

%// Get the first mxm block of kron-corresponding indices and then add to
%// each such block for the indices corresponding to the kron multiplications
%// of each iteration
a1 = bsxfun(@plus,reshape([0:N-1]*N+1,m,m),permute([0:N-1],[1 3 2]))

%// Now, a1 is a 3D array, we need to make 2D array out of it.
%// So, concatenate along rows to make it a "slimish" 2D array
a2 = reshape(permute(a1,[1 3 2]),size(a1,1)*size(a1,3),[])

%// Cut after every N rows to make it a square 2D array.
%// These are the indices for each frontal tensor of kron muliplications
slice_idx = reshape(permute(reshape(a2,N,size(a2,1)/N,[]),[1 3 2]),N,N)

%// -------------  PART 2: Get kron equivalent output

%// Perform x:(Nx1) x y:(1xN) multiplications
vals = bsxfun(@times,reshape(x,m*n,1,r),reshape(y,1,m*n,r)) %//multiplications

%// Get indices for all 3D slices and then index into those multiplications
%// with these for the final kron equivalent output
all_idx=bsxfun(@plus,slice_idx,permute([0:r-1]*m*m*n*n,[1 3 2])) %//all indices
out = vals(all_idx) %// final output of kron equivalent multiplications

两个张量之间的克罗内克积

Kronecker product between two tensors

matlab

matrix

vectorization

multiplication