[gowld]

The problem with "almost surely injective" for "practical purposes". Is that when you try to invert something, how do you know the result you get is one of those "practical purposes" ?

We aren't just trying to claim that two inputs are the same, as in hashing. We are trying to recover lost inputs.

[gnfargbl]

You don't, I guess. But again that's just the same as when you insert something into an object store: you can't be absolutely certain that a future retrieval will give you the same object and not a colliding blob. It's just good enough for all practical purposes.

[ajkjk]

Well that's not a problem, that's just a description of what "almost surely" means. The thesis is "contrary to popular opinion, you can more-or-less invert the model". Not exactly invert it--don't use it in court!--but like, mostly. The prevailing wisdom that you cannot is incorrect.