Question about tensor product between four-dimensional arrays-CodePudding

I'm trying to multiply together some 4 dimensional arrays (block matrices) in the following way:

where C has shape (50,50,12,6), Q has shape (50,50,12,12), R has shape (50,50,6,6),

I wonder how I should choose the correct axes to carry out tensor products? I tried doing matrix product in the following way:

H = np.tensordot(C_block.T,Q_block) @ C_block

But a value error is returned:

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
~\AppData\Local\Temp/ipykernel_2976/3668270968.py in <module>
----> 1 H = np.tensordot(C_block.T,Q_block) @ C_block

ValueError: operands could not be broadcast together with remapped shapes [original->remapped]: (6,12,12,12)->(6,12,newaxis,newaxis) (50,50,12,6)->(50,50,newaxis,newaxis)  and requested shape (12,6)

CodePudding user response：

Based on the sizes of your arrays, it looks like you are trying to do a regular matrix multiply on 50x50 arrays of matrices.

CT = np.swapaxes(C, 2, 3)
H = np.dot(np.dot(CT, Q), C)   R

CodePudding user response：

Creating some arrays of the right shape mix - with 10 instead of 50 for the batch dimensions. That is, treating the first 2 dimensions as batch that is repeated across all arrays including the result.

The sum-of-products dimension is size 12, right and left for Q.

This is most easily expressed with einsum.

In [71]: N=10; C=np.ones((N,N,12,6));  Q=np.ones((N,N,12,12)); R=np.ones((N,N,6,6))

In [73]: res = np.einsum('ijkl,ijkm,ijmn->ijln',C,Q,C) R

In [74]: res.shape
Out[74]: (10, 10, 6, 6)

dot does not handle 'batches' right, hence the memory error in the other answer. np.matmul does though.

In [75]: res1 = C.transpose(0,1,3,2)@Q@C   R

In [76]: res1.shape
Out[76]: (10, 10, 6, 6)

With all ones, the value test isn't very diagnostic, still:

In [77]: np.allclose(res,res1)
Out[77]: True

matmul/@ treats the first 2 dimensions as 'batch', and does a dot on the last 2. C.T in the equation should just swap the last 2 dimensions, not all.