Calculating l, the maximal number of simultaneously active features, yields strange results. For example, if we have 100 features and 100 neurons, l has to be < 100/(8 * ln(100)) = 2.7. But I would expect that 100 features can be simultaneously active because we have 100 dimensions, so the features can be orthogonal and independent. Am I understanding something wrong?
The math in the post is super hand-wavey, so I don’t expect the result to be exactly correct. However in your example, l up to 100 should be ok, since there is no super position. 2.7 is almost 2 orders of magnitude off, which is not great.
Looking into what is going on: I’m basing my results on the Johnson–Lindenstrauss lemma, which gives an upper bound on the interference. In the post I’m assuming that the actual interference is order of magnitude the same as the this upper bound. This assumption is clearly fails in your example since the interference between features is zero, and nothing is the same order of magnitude as zero.
I might try to do the math more carefully, unless someone else gets there first. No promises though.
I expect that my qualitative claims will still hold. This is based on more than the math, but math seemed easier to write down. I think it would be worth doing the math properly, both to confirm my claims, and it may be useful to have more more accurate quantitative formulas. I might do this if I got some spare time, but no promises.
my qualitative claims = my claims about what types of things the network is trading away when using super position
quantitative formulas = how much of these things are traded away for what amount of superposition.
Calculating l, the maximal number of simultaneously active features, yields strange results. For example, if we have 100 features and 100 neurons, l has to be < 100/(8 * ln(100)) = 2.7. But I would expect that 100 features can be simultaneously active because we have 100 dimensions, so the features can be orthogonal and independent. Am I understanding something wrong?
The math in the post is super hand-wavey, so I don’t expect the result to be exactly correct. However in your example, l up to 100 should be ok, since there is no super position. 2.7 is almost 2 orders of magnitude off, which is not great.
Looking into what is going on: I’m basing my results on the Johnson–Lindenstrauss lemma, which gives an upper bound on the interference. In the post I’m assuming that the actual interference is order of magnitude the same as the this upper bound. This assumption is clearly fails in your example since the interference between features is zero, and nothing is the same order of magnitude as zero.
I might try to do the math more carefully, unless someone else gets there first. No promises though.
I expect that my qualitative claims will still hold. This is based on more than the math, but math seemed easier to write down. I think it would be worth doing the math properly, both to confirm my claims, and it may be useful to have more more accurate quantitative formulas. I might do this if I got some spare time, but no promises.
my qualitative claims = my claims about what types of things the network is trading away when using super position
quantitative formulas = how much of these things are traded away for what amount of superposition.