I’ve been confused about the “defense-in-depth” cheese analogy. The analogy works in two dimensions, and we can visualize that constructing multiple barriers with holes will block any path from a point out of a three-dimensional sphere.
(What follows is me trying to think through the mathematics, but I lack most of the knowledge to evaluate it properly. Johnson-Lindenstrauss may be involved in solving this? (it’s not, GPT-5 informs me))
But plans in the the real world real world are very high-dimensional, right? So we’re imagining a point p (let’s say at (0,0,…,0)) in a high-dimensional space (let’s say Rn for large n, as an example), and an n-sphere around that point. Our goal is that there is no straight path from p to somewhere outside the sphere. Our possible actions are that we can block off sub-spaces within the sphere, or construct n-dimensional barriers with “holes”, inside the sphere, to prevent any such straight paths. Do we know the scaling properties of how many of such barriers we have to create, given such-and-such “moves” with some number of dimensions/porosity?
My purely guessed intuition is that, at least if you’re given porous n−1-dimensional “sheets” you can place inside of the n-sphere, that you need ≈exp(n) many of them with increasing dimensionality n. Nevermind, I was confused about this.
I’ve been confused about the “defense-in-depth” cheese analogy. The analogy works in two dimensions, and we can visualize that constructing multiple barriers with holes will block any path from a point out of a three-dimensional sphere.
(What follows is me trying to think through the mathematics, but I lack most of the knowledge to evaluate it properly.
Johnson-Lindenstrauss may be involved in solving this?(it’s not, GPT-5 informs me))But plans in the the real world real world are very high-dimensional, right? So we’re imagining a point p (let’s say at (0,0,…,0)) in a high-dimensional space (let’s say Rn for large n, as an example), and an n-sphere around that point. Our goal is that there is no straight path from p to somewhere outside the sphere. Our possible actions are that we can block off sub-spaces within the sphere, or construct n-dimensional barriers with “holes”, inside the sphere, to prevent any such straight paths. Do we know the scaling properties of how many of such barriers we have to create, given such-and-such “moves” with some number of dimensions/porosity?
My purely guessed intuition is that, at least if you’re given porous n−1-dimensional “sheets” you can place inside of the n-sphere, that you need ≈exp(n) many of them with increasing dimensionality n.Nevermind, I was confused about this.