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I read about PaCMAP dimensionality reduction method (PaCMAP).

They wrote that this method preserving both local and global structure of the data in original space.

  1. What is the meaning of preserving local structure of the data ?
  2. What is the meaning of preserving global structure of the data ?

Can you please add examples for those preserving structures ?

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When reducing the dimensionality of data, you want to keep local characteristics of data such is nearest neighbours while preserving the holistic approach such as keep far data points still far from each others. This is a trade-off that most dimensionality reduction algorithms, specially non-linear ones, try to keep. It is a trade-off because usually and naturally going toward one of them will destroy the other one, so what we want is preserving local structure while preserving the global one.

Example

See the nonlinear curve bellow. It looks like an English S. enter image description here

Now let's discuss modelling of two different distances here. Those distances are the distance between dark blue region at the beginning of S and:

  1. yellow region immediately under it which is on the curvature of S
  2. very light blue region which is on the curvature of the opposite side

How do you want to model it?

The global structure says that distance (1) is smaller than (2) (right?), but you cognitively see and know that the continues form of S shape suggests that distance (1) is actually larger than (2). Simply because you see the global structure of S and you see that this structure is a continues form of many local structures shown by different colours.

You intuitively know that if you walk on the data, you arrive to light blue region faster than yellow region! Here you are preserving local structure and if you don't, you will fall in the trap of seeing yellow region closer than blue one.

This is what LLE showing you. It embeds data in a lower dimensional space in which light blue is more close to dark blue rather than yellow region, that means local structure was safely preserved, while the global structure of the shape is reduced from a S shape to a simple band (see it as opening or flattening S-shape).

Hope I did not confuse you even more! Good luck

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    $\begingroup$ In your 3 dim reduction examples we see that just the local structure was preserved and not the global structure (distance 1 was not preserved), am I right ? $\endgroup$ Commented Mar 24, 2022 at 3:21
  • $\begingroup$ Thanks. Can you please add another example which shows that global structure is preserved ? (the distance between yellow region and dark blue is smaller than the distance between dark blue and light blue region ) ? $\endgroup$
    – Boom
    Commented Mar 24, 2022 at 3:32

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