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    $\begingroup$ I wouldn't expect that to hold for anything - after all, people use autoencoder latent spaces to have more meaningful similarity - for example, if you train a conv autoencoder on images, similar latent vectors correspond to semantically related images, as opposed to images similar in pixel space. $\endgroup$ Commented Jun 15, 2018 at 21:21
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    $\begingroup$ When you say "preserve (pairwise) distances", do you mean between all points of the input space, or just between the points of the training set? In other words, suppose I get a new input point $P$ (test point) which was not used at training time. Do you expect the autoencoder to learn a representation $Z$ such that the distance of $Z$ from any point in the reduced space, is the same distance as the distance of $P$ in the original space? $\endgroup$ Commented Jun 16, 2018 at 8:40
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    $\begingroup$ Or would you be content with an autoencoder which just learns representations of the training set which preserv the pairwise distances of the points in the training set? I.e., something like Multidimensional Scaling? $\endgroup$ Commented Jun 16, 2018 at 13:45
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    $\begingroup$ @DeltaIV The property I would like to see is that if a sample in the test set is close to a point in training/test set in the original space, it would be close in the reduced space as well. $\endgroup$ Commented Jun 18, 2018 at 6:14
  • $\begingroup$ Thank you very much. Please include this information in the body of the question: all important info should be there. Comments are meant to be temporary. $\endgroup$ Commented Jun 18, 2018 at 9:54