Translating np.einsum() to MATLAB - matlab

Question

I am having trouble understanding the documentation of np.einsum(). How are subscripts interpreted?
I am trying to write np.einsum('a...c,b...c', Y, conj(Y)) where Y is a matrix of shape C, F, T on the original python. Also, due to previous implementation differences my MATLAB Y is of size [F, T, C].
What does 'a...c,b...c' index in each component? I am confused.
How do I write the same instructions in MATLAB?

Answer 1

Quoting from the einsum documentation page:
To enable and control broadcasting, use an ellipsis. Default NumPy-style broadcasting is done by adding an ellipsis to the left of each term, like np.einsum('...ii->...i', a). To take the trace along the first and last axes, you can do np.einsum('i...i', a), or to do a matrix-matrix product with the left-most indices instead of rightmost, one can do np.einsum('ij...,jk...->ik...', a, b).
Later, an example is given:
>>> a = np.arange(25).reshape(5,5)
>>> np.einsum('...j->...', a)
array([ 10, 35, 60, 85, 110])
The equivalent MATLAB code to this example would be:
>> a = reshape(0:24, [5,5]).';
>> sum(a,2).'
ans =
10 35 60 85 110
Several things to note:
The ellipsis operator (...) should not be understood as "range", but as "whatever needs to be there".
"Broadcasting" refers to automatic replication of an array along the relevant dimension so that the mathematical operation is defined. This is a feature that exists in MATLAB since R2016b (called "implicit expansion").
You may notice several transpose operations (.') in the MATLAB equivalent. This is because numpy arrays are row-major while MATLAB array are column-major. Practically, while the underlying data has the same sequential order, a numpy array appears transposed compared to MATLAB. The transpositions were done so that arrays appear the same in intermediate stages.
Another example from these docs is:
>>> a = np.arange(6).reshape((3,2))
>>> b = np.arange(12).reshape((4,3))
>>> np.einsum('ki,jk->ij', a, b)
array([[10, 28, 46, 64],
[13, 40, 67, 94]])
>>> np.einsum('ki,...k->i...', a, b)
array([[10, 28, 46, 64],
[13, 40, 67, 94]])
>>> np.einsum('k...,jk', a, b)
array([[10, 28, 46, 64],
[13, 40, 67, 94]])
Which can be written as follows in MATLAB:
A = reshape(0:5, [2 3]).';
B = reshape(0:11, [3 4]).';
A.' * B.'
permute(sum( permute(A, [3 1 2]) .* B,2), [3 1 2])
shiftdim(sum( shiftdim(A, -1) .* B, 2), 2)
Several things to note:
When going from np.einsum('ki,jk->ij', a, b) to np.einsum('ki,...k->i...', a, b) you can see that the j-th dimension is replaced with .... The fact that both these examples have a -> in them, means it's in explicit mode.
When going from np.einsum('ki,jk->ij', a, b) to np.einsum('k...,jk', a, b), you can see that now the i-th dimension is replaced with .... The omission of ->...j simply demonstrates implicit mode (where the output dimensions are ordered alphabetically).