DavidHolmes
Hi Zack,
Can you clarify something? In the picture you draw, there is a codimension-1 linear subspace separating the parameter space into two halves, with all red points to one side, and all blue points to the other. Projecting onto any 1-dimensional subspace orthogonal to this (there is a unique one through the origin) will thus yield a `variable’ which cleanly separates the two points into the red and blue categories. So in the illustrated example, it looks just like a problem of bad coordinate choice.
On the other hand, one can easily have much more pathological situations; for examples, the red points could all lie inside a certain sphere, and the blue points outside it. Then no choice of linear coordinates will illustrate this, and one has to use more advanced analysis techniques to pick up on it (e.g. persistent homology).
So, to my vague question: do you have only the first situation in mind, or are you also considering the general case, but made the illustrated example extra-simple?
Perhaps this is clarified by your numerical example, I’m afraid I’ve not checked.
Thanks for the reply, Zack.
Sorry, I hope I didn’t suggest I thought that! You make a good point about some variables being more natural in given applications. I think it’s good to keep in mind that sometimes it’s just a matter of coordinate choice, and other times the points may be separated but not in a linear way.