The "continuous-looking" nature of the optimal 312-avoiding permanents we are finding suggests that we may want to put this way of approaching the problem in competition with a more traditional way. Namely, you could phrase the problem as learning a function from [1,..N] x [1,..N] where you impose the following (inconsistent!) "observations": you want f(x_1,y_1)f(x_2,y_2)f(x_3,y_3) = 0 whenever the three coordinates are in 312 formation, and you want f(x,y) = 1 for all (x,y). And then just in the traditional way try to learn some function R^2 -> R with a bunch of layers that satisfies these conditions as well as possible. I would guess this would do worse than what we're doing, but it might be nice to emphasize that we're beating other more traditional NN approaches.
(PS please let me know if "issue" is the wrong way to add comments/notes for the group; I am not fully git-literate!)
The "continuous-looking" nature of the optimal 312-avoiding permanents we are finding suggests that we may want to put this way of approaching the problem in competition with a more traditional way. Namely, you could phrase the problem as learning a function from [1,..N] x [1,..N] where you impose the following (inconsistent!) "observations": you want f(x_1,y_1)f(x_2,y_2)f(x_3,y_3) = 0 whenever the three coordinates are in 312 formation, and you want f(x,y) = 1 for all (x,y). And then just in the traditional way try to learn some function R^2 -> R with a bunch of layers that satisfies these conditions as well as possible. I would guess this would do worse than what we're doing, but it might be nice to emphasize that we're beating other more traditional NN approaches.
(PS please let me know if "issue" is the wrong way to add comments/notes for the group; I am not fully git-literate!)