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I'm currently trying to wrap my head around the t-SNE math. Unfortunately, there is still one question I can't answer satisfactorily: What is the actual meaning of the axes in a t-SNE graph? If I were to give a presentation on this topic or include it in any publication: How would I label the axes appropriately?

P.S: I read this Reddit question but the answers given there (such as "it depends on interpretation and domain knowledge"), don't really help me to understand this.

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Individual axes in t-SNE have no meaning at all.

Algorithms such as MDS, SNE, t-SNE, etc. only care about pairwise distances between points. They try to position the points on a plane such that the pairwise distances between them would minimize a certain criterion. This means that if you take a t-SNE plot and rotate it, then the resulting arrangement will be equally good as far as t-SNE is concerned. So the overall rotation that you get out of t-SNE algorithm is arbitrary.

To label the axes, I recommend writing something like "t-SNE dimension 1" and "t-SNE dimension 2".

(Sometimes people write "t-SNE 1" and "t-SNE 2" or some such, which is sloppy. Sometimes I see "t-SNE component 1" and "t-SNE component 2"; but I think that the word "component" does not work very well in this context.)

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    $\begingroup$ A corollary to this answer is that the Reddit suggestion is a little wrong. There probably isn't a reasonable way to interpret the axes in general, even if you have domain knowledge to apply. $\endgroup$ – shadowtalker May 30 '18 at 14:21
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    $\begingroup$ Another corollary to this answer is that the axes should be displayed on the same scale, so that if you multiplied by a rotator matrix and tilted your head correspondingly, the plot would look exactly the same. This is sadly far from standard practice (grumble!). $\endgroup$ – eric_kernfeld May 30 '18 at 15:17

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