# Wiki Contributions

This google search seems to turn up some interesting articles (like maybe this one, though I've just started reading it).

Paul [Christiano] called this “problems of the interior” somewhere

Since it's slightly hard to find: Paul references it here (ctrl+f for "interior") and links to this source (once again ctrl+f for "interior").  Paul also refers to it in this post.  The term is actually "position of the interior" and apparently comes from military strategist Carl von Clausewitz.

Can you clarify what figure 1 and figure 2 are showing?

I took the text description before figure 1 to mean {score on column after finetuning on 200 from row then 10 from column} - {score on column after finetuning on 10 from column}.  But then the text right after says "Babbage fine-tuned on addition gets 27% accuracy on the multiplication dataset" which seems like a different thing.

Vivek Hebbar2moΩ112

Note: The survey took me 20 mins (but also note selection effects on leaving this comment)

Vivek Hebbar3moΩ110

Here's a fun thing I noticed:

There are 16 boolean functions of two variables.  Now consider an embedding that maps each of the four pairs {(A=true, B=true), (A=true, B=false), ...} to a point in 2d space.  For any such embedding, at most 14 of the 16 functions will be representable with a linear decision boundary.

For the "default" embedding (x=A, y=B), xor and its complement are the two excluded functions.  If we rearrange the points such that xor is linearly represented, we always lose some other function (and its complement).  In fact, there are 7 meaningfully distinct colinearity-free embeddings, each of which excludes a different pair of functions.[1]

I wonder how this situation scales for higher dimensions and variable counts.  It would also make sense to consider sparse features (which allow superposition to get good average performance).

1. ^

The one unexcludable pair is ("always true", "always false").

These are the seven embeddings:

Vivek Hebbar3moΩ110

Oops, I misunderstood what you meant by unimodality earlier. Your comment seems broadly correct now (except for the variance thing). I would still guess that unimodality isn't precisely the right well-behavedness desideratum, but I retract the "directionally wrong".

Vivek Hebbar3moΩ230

The variance of the multivariate uniform distribution  is largest along the direction , which is exactly the direction which we would want to represent a AND b.

The variance is actually the same in all directions.  One can sanity-check by integration that the variance is 1/12 both along the axis and along the diagonal.

In fact, there's nothing special about the uniform distribution here: The variance should be independent of direction for any N-dimensional joint distribution where the N constituent distributions are independent and have equal variance.[1]

The diagram in the post showing that "and" is linearly represented works if the features are represented discretely (so that there are exactly 4 points for 2 binary features, instead of a distribution for each combination).  As soon as you start defining features with thresholds like DanielVarga did, the argument stops going through in general, and the claim can become false.

The stuff about unimodality doesn't seem relevant to me, and in fact seems directionally wrong.

1. ^

I have a not-fully-verbalized proof which I don't have time to write out

Vivek Hebbar4moΩ110

Maybe models track which features are basic and enforce that these features be more salient

Couldn't it just write derivative features more weakly, and therefore not need any tracking mechanism other than the magnitude itself?