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I have a set of objects u1, u2,..., un and an algorithm which gives a R(1000) [1000 dimensional] vector for each object.

Table 1
   t1  t2  ... t1000
u1 234 157 ... 92
u2 117 157 ... 39
...
un 234 157 ... 39

From this data, I'm calculating hamming distance between these objects:

Table 2
row col distance
u1  u2  200
u1  u3  500
u4  u2  200
u4  u3  900
...

I want to create a 2D plot in which each object is a point and distance between points comes from Table 2.

For that I need absolute positions of these objects in some 2D space. Is there a way to find or simulate absolute positions when only distances are known. If yes, how do I select what to put on x-axis and y-axis?

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  • 1
    $\begingroup$ If the points are in 1000 dimensions, there's no reason to think you can represent them in two. $\endgroup$ – Gerry Myerson Jun 28 at 9:30
  • $\begingroup$ @Milloupe And satisfy $n\times n$ constraints for arbitrary $n$? $\endgroup$ – broncoAbierto Jun 28 at 9:47
  • $\begingroup$ Unless the $n$ points are somehow coplanar. (dimensionality reduction?) $\endgroup$ – peterwhy Jun 28 at 9:51
  • $\begingroup$ @peterwhy points are not coplanar. If I do dimensionality reduction, the distance between the points changes, both order and magnitude. $\endgroup$ – penguin Jun 28 at 11:07
  • $\begingroup$ @Milloupe The distance in this case is hamming distance. How is it possible to plot all points in 1D? Take this example, 1 and 2 are 100 places apart, 2 and 3 are 500 places apart, 3 and 1 are 50 places apart. How do I place these points in 1 dimension? $\endgroup$ – penguin Jun 28 at 11:15
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Figured out a way to this - Multidimensional Scaling https://en.wikipedia.org/wiki/Multidimensional_scaling

This may work accurately when "euclidean" distances are known. Not sure how much information does it compromise with hamming distances.

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