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Suppose I have two tensors $A_{i_1,\ldots,i_M}$ and $B_{j_1,\ldots,j_N}$ where $M \neq N$ in general. We can define a tensor $C_{i_1,\ldots,i_M,j_1,\ldots,j_N}$ by

$$ C_{i_1,\ldots,i_M,j_1,\ldots,j_N} = A_{i_1,\ldots,i_M} + B_{j_1,\ldots,j_N} $$

I thought I could use torch.einsum to construct such a tensor in pytorch but it doesn't do the job unfortunately. Is there a way to do this in pytorch?

I wonder mostly if there's a pytorch API that can do this rather than me working out several tricks here and there.

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I would tackle your problem as follows:

  1. Compute rank-$(M+N)$ tensors $\tilde{A}$ and $\tilde{B}$ that satisfy: $$ \forall j_1, ..., j_N\, \forall i_1, ..., i_M: \quad \tilde{A}_{i_1, ..., i_M, j_1, ..., j_N} = A_{i_1, ..., i_M}, \\ \forall i_1, ..., i_M\, \forall j_1, ..., j_N: \quad \tilde{B}_{i_1, ..., i_M, j_1, ..., j_N} = B_{j_1, ..., j_N}. $$

  2. Compute the answer by $$ C = \tilde{A} + \tilde{B}. $$ The second step is trivial and thus you just need to figure out how to do the first step. Computing the tensors $\tilde{A}$ and $\tilde{B}$ could possibly be done by expanding the dimensions to be $M+N$ and then stacking along the new dimensions or using torch.repeat_interleave. My answer is slightly different since you can also compute these larger tensors via outer products of $A$ and $B$ with tensors that contain only ones, i.e. $$ \tilde{A}_{i_1, ..., i_M, j_1, ..., j_N} = A_{i_1, ..., i_M} \times 1_{j_1, ..., j_N}, \\ \tilde{B}_{i_1, ..., i_M, j_1, ..., j_N} = 1_{i_1, ..., i_M} \times B_{j_1, ..., j_N}. $$ Here, is some PyTorch code that works for $M = 3$ and $N = 2$, but could also be modified for other values of $M$ and $N$:

    1. A_new = torch.einsum("ijk,lm->ijklm", A, torch.ones_like(B))

    2. B_new = torch.einsum("ijk,lm->ijklm", torch.ones_like(A), B)

    3. C = A_new + B_new

For more PyTorch API specific questions, you can also check out their official forum: https://discuss.pytorch.org/

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    $\begingroup$ The meaning of your equalities in point 1 is not clear. Can you please clarify your definitions? You have N+M indices on the lhs but only M or N on the rhs $\endgroup$ Commented Feb 23 at 22:47
  • $\begingroup$ @user8469759 Basically $\tilde{A}$ is just $A$ repeated along the new right $N$ dimensions and $\tilde{B}$ is just $B$ repeated along the new left $M$ dimensions. $\endgroup$ Commented Feb 24 at 14:22

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