That’s good to hear! And I agree with your new intuition.
I think if you want interference terms to actually be zero you have to end up with tegum products, because that means you want orthogonal vectors and that implies disjoint subspaces. Right?
I don’t think so? If you have eg 8 vectors arranged evenly in a 2D plane (so at 45 degrees to each other) there’s a lot of orthogonality, but no tegum product. I think the key weirdness of a tegum product is that it’s a partition, where every pair in different bits of the partition is orthogonal. I could totally imagine that eg the best way to fit 2n vectors is n dimensional space is two sets of n orthogonal vectors, but at some arbitrary angle to each other.
I can believe that tegum products are the right way to maximise the number of orthogonal pairs, though that still feels a bit weird to me. (technically, I think that the optimal way to fit kn vectors in R^n is to have n orthogonal directions and k vectors along each direction, maybe with different magnitudes—which is a tegum product. It forming 2D-3D subspaces feels odd though).
I think partitions can get you more orthogonality than your specific example of overlapping orthogonal sets. Take n vectors and pack them into d dimensions in two ways:
A tegum product with k subspaces, giving (n/k) vectors per subspace and n^2*(1-1/k)orthogonal pairs.
(n/d) sets of vectors, each internally orthogonal but each overlapping with the others, giving n*d orthogonal pairs.
If d < n*(1-1/k) the tegum product buys you more orthogonal pairs. If n > d then picking large k (so low-dimensional spaces) makes the tegum product preferred.
This doesn’t mean there isn’t some other arrangement that does better though...
That’s good to hear! And I agree with your new intuition.
I think if you want interference terms to actually be zero you have to end up with tegum products, because that means you want orthogonal vectors and that implies disjoint subspaces. Right?
I don’t think so? If you have eg 8 vectors arranged evenly in a 2D plane (so at 45 degrees to each other) there’s a lot of orthogonality, but no tegum product. I think the key weirdness of a tegum product is that it’s a partition, where every pair in different bits of the partition is orthogonal. I could totally imagine that eg the best way to fit 2n vectors is n dimensional space is two sets of n orthogonal vectors, but at some arbitrary angle to each other.
I can believe that tegum products are the right way to maximise the number of orthogonal pairs, though that still feels a bit weird to me. (technically, I think that the optimal way to fit kn vectors in R^n is to have n orthogonal directions and k vectors along each direction, maybe with different magnitudes—which is a tegum product. It forming 2D-3D subspaces feels odd though).
Oh yes you’re totally right.
I think partitions can get you more orthogonality than your specific example of overlapping orthogonal sets. Take n vectors and pack them into d dimensions in two ways:
A tegum product with k subspaces, giving (n/k) vectors per subspace and n^2*(1-1/k)orthogonal pairs.
(n/d) sets of vectors, each internally orthogonal but each overlapping with the others, giving n*d orthogonal pairs.
If d < n*(1-1/k) the tegum product buys you more orthogonal pairs. If n > d then picking large k (so low-dimensional spaces) makes the tegum product preferred.
This doesn’t mean there isn’t some other arrangement that does better though...
Yeah, agreed that’s not an optimal arrangement, that was just a proof of concept for ’non tegum things can get a lot of orthogonality