Neural network doesn't actually mean machine learning, it only means machine learning if you do machine learning with it (i.e. optimisation with data). Otherwise it's just a general function approximator.

I don't see why you need any machine learning at all, this is a pure optimisation problem. ML happens when optimisation is based on data of some kind. That's why an objective function like MSE is needed, because a well behaved loss function isn't known generally.

You don't really need the data at all since you have an objective to minimise for the optimal transform, you can just initialise an X, ideally a good guess, and optimise it for your objective function.