My question is: Is there any mathematical or empirical way to prove that given a dataset containing noisy signals y(t) [Y = X +N] and another dataset containing noise N and we want generator to generate clean signals X ̂ . How to prove other than experiments that generator will be able to generate clean signals from random noise vector z.
You could look at Shannon's theory on information entropy (see Shannon, 1948 for original work). If you can quantify the data you are transmitting [e.g. bits or bytes of data and by type], along with the noise you should be able identify if your signal generator can theoretical produce a clean signal.
You want to prove that your GAN algorithm is optimal filter. Usually this is done by showing that it can Minimize the Mean Square Error. You want to show that you algorithm minimizes the variance in the estimated states.
You might find some examples by googling proof optimal filter
Thank you so much for your answers.
Janez Podobnik I have been working since your answer and found the optimality condition and also found the proofs of optimal estimator. I have minimized the summary as you can see here in the attached picture.
But I am stuck in linking the attached theory with my case. Since this is the general condition for an optimal estimator (attached picture) and yes generator would be minimizing a cost function but how can I say that my generator would be able to generate clean signals?
Also, from the optimal estimator Equation, we can say that the estimate of x would be a scaled version of y since varinaces are constant. Which means this is the best an estimator can estimate. how this estimator fits into generator theory.
I have changed the notations in main question for simplicity purpose.
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