The idea the neighbors of my neighbors are likely to be neighbors is basically an assumption of positive curvature. In large codimension embdedded submanifolds can have very negative curvature, in which case neighbors of neighbors might not be as likely to be neighbors as one might first think.

I don't think that's a serious restriction.

Firstly - in general it's trivially true that you have probability 0 to cleanly embed a high dimensional space into a 2/3 dimensional representation over the set of all possible high dimensional data - yet interesting data often does have lower-dimensional structure.

Secondly - so what? Can you think of plausible scenario where this assumption does not hold and it's possible to generate a low-dimensional embedding? If it's impossible to embed, then it's not an algorithmic problem if you fail to find an embedding.

Correct. The authors in fact mention this in the paper, and state that this is probably not an issue because in practice most large high-dimensional datasets tend to have points which lie on/near embedded submanifolds of much lower dimension.

I created a throwaway just to pretend my salary isn't public record.

Location : Eugene, OR.

Position: Assistant Professor of Mathematics Tenure: 3 years Base pay (9 month) $72,000 in 2013, increased to about $75,000 in 2016. (minus union dues). Will increase to $82,000 when I make tenure.

Signing bonus: $0

Stock: $0

Pension : keeping my fingers crossed.

Experience when hire. Ph.D plus 5 years post-doc (at Princeton.)

Gender : M Native speaker: Yes.

This is exactly the reason I decided not to stay in academia after I got my Ph.D. I could not rationalize working 10 to 12 hours a day, 7 days a week, trying to teach, conduct research, scramble for funding, participate in department politics just to attain tenured position at an R1 school while making barely 75K starting.

I now work in a development position at a large tech company for which I'm overqualified. I work about 7 hours a day plus a couple of meetings. I never work weekends, have plenty of free time to conduct research on the side and make well into 6 figures.