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I am looking for ideas to find an $n\times n$ real matrix $X$ which approximately solves

$$X^{\otimes m} a = b$$

where $a$ and $b$ are given $n^m$-dimensional real vectors (or perhaps $n^m\times k$ matrices). There is lots of material on finding $a$ from $X$ and $b$, which is rather different.

If it is easier, though it probably isn't, I am also interested in cases where there might be a set of such equations, to be solved simultaneously for the same $X$, for example

$$X a_1 = b_1$$ $$(X\otimes X)\, a_2 = b_2$$ $$(X\otimes X \otimes X)\, a_3 = b_3.$$

Background: The problem comes from considering iterated-integral signatures in rough path theory, and trying to interpret a given truncated signature of an unknown path, asking which linear transformation of a certain known path makes it like the unknown one.

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This is a "procrustes problem". This 2003 paper by Bojanczyk and Lutoborski gives an approach when $m=2$.

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