# Reconstructing linear maps with particular properties using machine learning

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.

• 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/… – Popescu Claudiu Feb 1 at 18:13
• 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. – D.W. Feb 1 at 18:40