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I'm trying to gain an intuition for the 2nd dimension in the spectral embedding of an S-shaped dataset as in this example: enter image description here

The 1st dimension seems to neatly capture the local similarity between points, but is there a similar physical interpretation for the 2nd dimension? More specifically, why do both red AND blue points show low values for this dimension?

I am wondering whether these dimensions can be interpreted in a similar way as done for the embedding of hand-written digits:

enter image description here

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  • $\begingroup$ Consider the sample in the bottom left of the LLE. It's not thick at all. It didn't capture thickness on the y axis. That was just a pretty story to make it sound cool. $\endgroup$
    – Has QUIT--Anony-Mousse
    May 30, 2019 at 5:20

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The first dimension yields nthr axis along which to split to get two partitions.

The second dimension is for splitting into 3 or 4 clusters. Try running k-means on this projection with k=3 or k=4 and you should get some decent results.

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