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I sample a 2D surface in $\mathbb{R}^3$ with $N$ points, and compute an isomap using pairwise weighted geodesic distances. I am thus able to embed this surface into a $M$ dimensional space in which pairwise euclidean distances closely match their respective weighted geodesic distances on the original manifold. In brief, I just perform a standard isomap, solved with the SMACOF procedure.

However, I am wondering if this approach can lead to a self-intersecting manifold. If so, does increasing the dimensionality $M$ of the target space guarantee that at some point their won't be any intersection remaining ? Otherwise, how can I prevent self intersections from happening ?

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You can avoid self-intersections by working in a higher dimension. Take a look at the Whitney embedding theorem: en.wikipedia.org/wiki/Whitney_embedding_theorem –  Simon Lyons Aug 22 '11 at 20:11
    
thanks! But do you know if, in the specific case of isomaps, increasing the dimension would avoid self-intersections in the same way as it does with Whitney's trick ? –  WhitAngl Aug 22 '11 at 20:50
    
What's an isomap? –  Igor Rivin Aug 23 '11 at 6:43
    
it is a multidimensional scaling (MDS) using geodesic distances as input. And MDS is a technique which tries to recover a set of N dimensional points whose pairwise euclidean distances closely match a given distance matrix. If this given distance matrix is also euclidean, the result can be computed using a PCA, otherwise we can rely on the SMACOF technique which is an iterative procedure. –  WhitAngl Aug 23 '11 at 8:53
    
What is SMACOF? What is PCA? Why do you expect people to know this jargon? And by the way, MDS is a heuristic to solve an essentially intractable problem... –  Igor Rivin Aug 23 '11 at 14:55

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