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Consider the expression $AX = Y$ with $A$ an $N\times N$ complex matrix, while $X,Y$ are $N$-dimensional complex vectors. Let's further assume that $A$ is unitary, $A^{\dagger}A= I$. $A$ also satisfies some more technical assumptions that I don't think are necessary to get into here, but can explain if necessary; suffice to say that they make one's life difficult.

My goal is to develop a machine learning algorithm which, when given the input vector $X$ and output vector $Y$, can generate the matrix $A$ which relates them. While this may sound somewhat trivial, the difficult part is finding the $A$ which satisfies the technical assumptions which I've swept under the rug. My (admittedly vague) strategy would be to generate an ensemble of randomly generated operators $A$ and vectors $X,Y$, which can be done efficiently. It seems that one would want to then somehow train a neural network to best represent $A$ by minimizing the difference between the machine's prediction for $Y$ and the true $Y$. I have not run across this kind of problem in the machine learning literature before, but I assume it has been done in some guise, so any suggestions about existing architectures or problem types to consider would be very welcome.

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  • $\begingroup$ Since you want to use machine learing , I assume you have some data in the form of X , Y pairs . Is this right ? This paper may be useful : pdfs.semanticscholar.org/163c/… $\endgroup$ – Popescu Claudiu Feb 1 at 18:13
  • $\begingroup$ I suspect there might be a nice way to solve this directly with linear algebra, instead of using machine learning, similar to how we solve linear regression problems; but I lack the knowledge to work out whether that is indeed the case. $\endgroup$ – D.W. Feb 1 at 18:40

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