I'm afraid I was not talking about the same thing. The paper you quoted doesn't claim one can improve hearing or physical signal processing. When we say noise in science (not the same as a noisy environment) usually we mean no computer no matter how powerful can ever de-noise whatever the signal is hidden in. Let me illustrate;
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When you introduce a bias of two electrons into a 2D field measurement you can subtract it. When one introduces noise which is by definition random no series of action can ever remove it. When the noise is larger than the signal nothing useful can be recovered. In this case one could apply a kernel (contrast or brightness increase) to increase the contrast between the noise and the signal however when you don't have a 2D field, but just a 1D signal (auditory) you can't increase contrasts. There is no clever signal processing that could ever restore the signal.
We can train a rat or a human to "extract information from weak signals in noisy backgrounds" but the limits here are obvious. Firstly the sound must be heard in the first place and thus already above the noise levels, there is only a slight improvement (not a recovery), and "learning does not typically generalize to untrained stimuli" which means that this research is actually highlighting conditioning, not a signal processing improvement, hence it's pretty much useless.
My problem is that often I can not hear the entire sentence. I can't even hear myself breathe in a silent room because of my decently loud tinnitus. The general rule is that x/sqrt(x) is the amount of poisson noise in any signal x. If you mix a signal with some tinnitus (and it's noise) you're going to have more noise and so be less able to understand what is being said. Maybe conditioning can reduce the severity slightly by improving your 'perceptual salience', but the physical problem remains severe and the actual auditory system that is causing the problem is not being altered by the conditioning.