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simplifying expressions - Function to convert TensorContract[_TensorProduct, indices] into equivalent Dot + Tr version


Mathematica can use either Dot + Tr to represent some tensors, or TensorContract + TensorProduct. I believe that the TensorContract + TensorProduct representation, while verbose, is more powerful for a couple reasons:



  1. It can represent a wider variety of tensors, e.g., TensorContract[TensorProduct[a, b], {{1, 4}, {2, 5}, {3, 6}}] where a and b are rank 3 tensors doesn't have an equivalent Dot + Tr representation (at least, I can't think of one).

  2. TensorReduce can in some cases reduce pure TensorContract + TensorProduct expressions better than the equivalent Dot + Tr expressions.



Because of the above, it would be convenient to have a function that converted a Dot + Tr representation into a TensorContract + TensorProduct representation. Another reason why it would be nice to have such a function is that TensorReduce of a pure TensorContract + TensorProduct often works much better than TensorReduce of a mixture of a Dot + Tr and TensorContract + TensorProduct representation.


Pure vs mixed


Here is an example where TensorReduce works better with pure TensorContract representations instead of mixed representations:


TensorReduce[
    r.R - TensorContract[TensorProduct[R, r], {{1, 2}}],
    Assumptions -> (r|R) \[Element] Vectors[3]
]

TensorReduce[

    TensorContract[TensorProduct[r, R], {{1, 2}}] - TensorContract[TensorProduct[R, r], {{1, 2}}],
    Assumptions -> (r|R) \[Element] Vectors[3]
]


r.R - TensorContract[r[TensorProduct]R, {{1, 2}}]


0



ToTensor


The following function can be used to convert at Dot + Tr representation into a TensorContract + TensorProduct representation:



ToTensor[expr_] := expr /. {Dot->dot, Tr->tr}

dot[a__] := With[{indices = Accumulate@Map[TensorRank]@{a}},
    TensorContract[TensorProduct[a], {#, # + 1} & /@ Most[indices]]
]

tr[a_] /; TensorRank[a] == 2 := TensorContract[a, {{1, 2}}]
tr[a_, Plus, 2] := TensorContract[a, {{1, 2}}]
tr[a___] := Tr[a]


FromTensor


It would be nice to have a function that converts a TensorContract + TensorProduct representation into a Dot + Tr representation, if possible. Let's call such a function FromTensor. Then, a TensorSimplify function that does something like FromTensor @ TensorReduce @ ToTensor @ expr could be defined that is as powerful as a simple TensorReduce, but allows one to work with Dot + Tr or mixed representations.


Examples


The kinds of TensorContract + TensorProduct representations that should be converted into a Dot + Tr representation include at least the following, where a and b are vectors, and m and n are matrices:



  1. Tr[m.n] ⇔ TensorContract[TensorProduct[m, n], {{1, 4}, {2,3}}]

  2. m.n ⇔ TensorContract[TensorProduct[m, n], {{2, 3}}]

  3. a.m.n ⇔ TensorContract[TensorProduct[a, m, n], {{1, 2}, {3, 4}}]

  4. a.m.n.b ⇔ TensorContract[TensorProduct[a, m, n, b], {{1, 2}, {3, 4}, {5, 6}}]



Some other similar examples:



  1. a.Transpose[n].Transpose[m] ⇔ TensorContract[TensorProduct[a, m, n], {{1, 5}, {4, 3}}]

  2. Tr[Transpose[m].n] ⇔ TensorContract[TensorProduct[m, n], {{1, 3}, {2, 4}}]


There may be other equivalent representations.


So, my question is, can somebody write such a FromTensor function?


(I have written such a function, but I am unhappy with it. I'm hopeful that someone can write a better one. I will post my version as an answer at some point, but for now I'm curious what other independent answers are possible)




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