3 ms·
The idea the neighbors of my neighbors are likely to be neighbors is basically an assumption of positive curvature. In large codimension embdedded submanifolds
by eugene_ducker 10y ago
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.
- emn13 10y agoI 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.
- cs702 10y agoCorrect. 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.