Actually one more thing I’m probably also gonna do is create a big subspace overlap matrix and factor it in some way to see if I can split off some different modules. I had intended to do that originally, but the finding that all the dimensions were used at least half the time made me pessimistic about it. But I should Try Harder.
One thing I’m thinking is that the additive structure on its own isn’t going to be sufficient for this and I’m going to need to use intersections more.
Realization: the binary multiplicative structure can probably be recovered fairly well from the binary additive structure + unary eigendecomposition?
Let’s say you’ve got three subspaces X, Y and Z (represented as projection matrices). Imagine that one prompt uses dimensions D1=X+Y, and another prompt uses dimensions D2=Y+Z. If we take the difference, we get X−Z. Notably, the positive eigenvalues correspond to X, and the negative eigenvalues correspond to Z.
Define f(P) to yield the part of P with positive eigenvalues (which I suppose for projection matrices has a closed form of P+P22, but the point is it’s unary and therefore nicer to deal with mathematically). You get D1∧¬D2=f(D1−D2), and you get D1∧D2=D1−f(D1−D2).
Actually one more thing I’m probably also gonna do is create a big subspace overlap matrix and factor it in some way to see if I can split off some different modules. I had intended to do that originally, but the finding that all the dimensions were used at least half the time made me pessimistic about it. But I should Try Harder.
One thing I’m thinking is that the additive structure on its own isn’t going to be sufficient for this and I’m going to need to use intersections more.
Realization: the binary multiplicative structure can probably be recovered fairly well from the binary additive structure + unary eigendecomposition?
Let’s say you’ve got three subspaces X, Y and Z (represented as projection matrices). Imagine that one prompt uses dimensions D1=X+Y, and another prompt uses dimensions D2=Y+Z. If we take the difference, we get X−Z. Notably, the positive eigenvalues correspond to X, and the negative eigenvalues correspond to Z.
Define f(P) to yield the part of P with positive eigenvalues (which I suppose for projection matrices has a closed form of P+P22, but the point is it’s unary and therefore nicer to deal with mathematically). You get D1∧¬D2=f(D1−D2), and you get D1∧D2=D1−f(D1−D2).
Maybe I just need to do epic layers of eigendecomposition...