Previous stylisation was slow because it needed to SGD optimisation for each image to be stylised. This uses a NN trained once. When you've trained a NN it is precisely a closed form solution, in the style y = max(0, 3x + 4). However they are normally a little longer to wrote down :P
Ah okay right this is the answer. Previous approaches [1] are deep generative models that you have to optimize for each input, whereas here you run just a forward evaluation on a model that you've trained beforehand.
I would still argue the term closed-form is misleading here, because:
- Even during training at any given time you can read off a "closed-form expression" of the neural network of this type, so closed-form in this broad sense really doesn't mean much. Furthermore any result of any numerical computation ever are also closed-form solutions according to this, on the grounds that they result from a computation that completed in finite number of steps. So really whenever you ask a grad student to run some numerical simulation expect them to come back saying "Hey I found a closed-form expression!"
- The reason the above is absurd is that these trained NN's aren't really solutions to the optimization problem, but approximations. So this is really saying I have a problem, I don't know how to solve it but I can produce a infinite sequence of approximations. Now I'm gonna truncate this sequence of approximations, and call this a closed form solution.
The analogy in highschool math would be computing an infinite sum that doesn't converge, but now let's instead just add to some large N, and call this a closed-form solution.
Actually, I agree with you. Initially you seemed to object to the term "closed form"; this now highlights the more pertinent point - these models are 100% closed form, but 0% "solution" in the formal sense.
Someone correct me if I'm wrong, but I believe this refers to the fact that it can be expressed in terms of certain simple mathematical operations like addition, subtraction, multiplication, powers, roots etc.—and as a consequence, the execution is very efficient. My understanding is that 'closed form' solution is essentially something that resembles a polynomial (again, accepting corrections!).
Closed form just means you can do it in a finite number of operations. So just "run X" rather than the previous versions of this kind of thing which are "repeat X until measure Y is lower than the limit I care about". (my basic understanding)
I checked the Wikipedia article, and the sorts of operations involved do appear to be a part of the definition: https://en.wikipedia.org/wiki/Closed-form_expression —though it sounds like it's a somewhat loosely defined term.
