Dimensionality reduction: Locally Linear Embedding

One of the most cited algorithm in nonlinear manifold learning, with Isomap, is LLE. Contrary to Isomap, LLE tries to retain the local data structure of the sampled manifold. Whereas Isomap preserves absolute distances, LLE preserves local relative distances (it preserves barycenter weights).

This means that LLE is not suitable for every dimensionality reductions. For visualization purposes, it can lead to very different solutions if the manifold is noisy.

When compressing the SCurve manifold that was shown in my other tickets, LLE gives this result :
LLE compression of the SwissRoll

It is obvious that the local structure is preserved (the color gradients), but the global structure is not : the result should be some kind of rectangle, not a triangle. Other local techniques that preserve local distances (and not relative ones) can lead to better results, so it really is a “relative” problem.

Although the results are not good for visualization purposes, LLE is very fast to compute, and classification on this subspace is efficient, because of the preservation of the local structure.

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