Implement Einstein Sum in Causal Tensor

[marvin-hansen]

We all love Einstein...

However, conventionally, Einstein Sum over Tensors is implemented as string parsing and index matching. What could possibly go wrong when mixing a parser wth a calculator? Everything,
However, the new deep_causality_ast crate with its immutable ConstTree allows a different direction that basically implements the Einstein Sum Parameters as a fixed arithmetic tree that also ensures the configurtaiton is compiler checked.

Also, Einstein sum will be needed sooner rather than later to reduce complex TriTensors, so its about high time to get this done.

[marvin-hansen]

Fundamental Operations Expressible by einsum

Operation	Input/Tensor Type	einsum Logic (Conceptual)	Purpose/Context
1. Generalized Contraction	Two or more Tensors	$\text{'..., i, j, ...', '\ldots, j, k, \ldots' \rightarrow \text{'}\ldots, i, k, \ldots\text{'}}$	The core operation for Tensor Networks, Process Matrix (ICS) calculation, and multi-variable dependencies.
2. Matrix Multiplication (MM)	Two Rank-2 Tensors	$\text{'ij, jk } \rightarrow \text{ ik'}$	The most frequent operation in neural networks and linear systems (BLAS GEMM).
3. Dot Product (Vector-Vector)	Two Rank-1 Tensors	$\text{'i, i } \rightarrow \text{ ' '}$ (scalar output)	Simple inner product and basis projection.
4. Trace	Single Tensor (Rank $\ge 2$)	$\text{'ii } \rightarrow \text{ ' '}$ (scalar output)	Summation over the diagonal. Essential for the final collapse/measurement $\text{Tr}[\ldots]$ in quantum systems.
5. Tensor Product (Outer Product)	Two Tensors (Rank $\ge 1$)	$\text{'ij, kl } \rightarrow \text{ ijkl'}$ (no contraction/sum)	Combines two independent systems (e.g., combining two channels $\mathcal{E}_A \otimes \mathcal{E}_B$).
6. Element-wise Product (Hadamard)	Two Tensors (Matching Shape)	$\text{'ijk, ijk } \rightarrow \text{ ijk'}$ (no contraction/sum)	Direct multiplication of corresponding elements.
7. Transpose / Permutation	Single Tensor	$\text{'ijk } \rightarrow \text{ kji'}$ (no contraction/sum)	Reordering of axes. Crucial for optimizing contraction performance.
8. Diagonal Extraction	Single Tensor (Rank $\ge 2$)	$\text{'ii... } \rightarrow \text{ i...'}$	Extracts the diagonal elements of a higher-rank tensor into a lower-rank vector.
9. Summation / Reduction	Single Tensor	$\text{'ijk } \rightarrow \text{ ij'}$	Summing a tensor over specified axes (e.g., reducing a volume over the $k$ axis).
10. Batch Matrix Multiplication	Two Rank-3+ Tensors	$\text{'b ij, b jk } \rightarrow \text{ b ik'}$	Performing many matrix multiplications in parallel (the b index is the batch dimension).

[aaelony]

this is awesome. I have something similar from a while back; https://github.com/aaelony/yet_another_einsum_playground/blob/main/src/main-playground.py ... would be great to have an einsum! macro or something like that (if it doesn't already exist).

[marvin-hansen]

@aaelony ein_sum has been added in PR #375 and was merged today.

[marvin-hansen]

Code examples: https://github.com/deepcausality-rs/deep_causality/blob/main/deep_causality_tensor/examples/ein_sum_causal_tensor.rs