I don't mean to be super negative, but because of the general tone early in the article and some sloppy notation, I never finished reading. I think the goal of

I don't mean to be super negative, but because of the general tone early in the article and some sloppy notation, I never finished reading. I think the goal of an article like this should be to give a high-level intuitive explanation for some technical result, rather than sounding smart or complicated. First, it is a little weird to me to talk about "old-school ML" as learning maps from inputs to hidden features. That seems neither old, nor very representative of the field of Machine Learning as a whole. It's also weird to say that RBMs and other deep learning algorithms are formulated using classical statistical mechanics. Moreover, implying that this scary-sounding formulation is the reason they are interesting seems like an attempt at sounding smart. Typically there are many ways to motivate and derive different algorithms, and it is /useful/ to acknowledge the multiple viewpoints because they often give different insights. Second, the section about flow maps and fixed points seems to make a mess out of the notation by either being unclear or disagreeing with standard notation. What is meant by the notation "f(X) -> X"? Presumably this means something like f is a function that maps elements of the set X to elements in the set X. More standard notation for this would be something like "f: X -> X". Perhaps it means that the image of the set X under the function f is again the set X. But does that require that f be a surjective function? Confusingly, it also looks like the function f might be required to be the identity function, but given the context this is clearly not the intended interpretation. When defining the fixed point, it seems that it would be more natural to say that x is a fixed point of f if f(x) = x. That is, x is fixed or unmoved by the function f. It turns out that for contractions (and some other functions, too), that the sequence f(x), f(f(x)), f(f(f(x))), and so on is guaranteed to converge to a unique fixed point of f. The notation f^n typically refers to the function f being applied n times, which is not the usage in the article. In the article, f^1, f^2, and so on are all identical copies of the function f. Using the standard notation, the definition of f_infty would be f_infty(x) = lim_{n -> infty} f^n(x). And, in the case of a contraction, the Banach fixed point theorem gives that f_infty is well-defined, and there exists a unique x_fix in X so that f_infty(x) = x_fix for all x in X (i.e., iterating f repeatedly converges to a unique fixed point x_fix of the function f). These things do not necessarily mean that the article is uninteresting or uninformative or even technically incorrect. But if the author didn't take the time to make the simple things clear, then I'm not sure that I want to read the rest. Sorry for the rant.