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I am trying to learn data in higher space into lower space. To have a clue, I'd like to know how to transform the data in the image below into a lower dimension preserving the structure. Hope to hear some explanations and what should I study to learn mapping data to lower dimension?

enter image description here

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  • $\begingroup$ Search for methods of dimensionality reduction such as PCA or LLE. Start with PCA. $\endgroup$ – Nikolas Rieble Feb 13 '17 at 16:40
  • $\begingroup$ What is LLL short for? $\endgroup$ – user122358 Feb 13 '17 at 16:42
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    $\begingroup$ stats.stackexchange.com/questions/2691/… is a good starting point $\endgroup$ – Sycorax Feb 13 '17 at 16:42
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    $\begingroup$ PCA is one possibility, but it does not explicitly aim at preserving distances when mapping data to lower dimensions, as MDS does. $\endgroup$ – Stephan Kolassa Feb 13 '17 at 16:43
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    $\begingroup$ There are various dimensionality reduction techniques out there. It all depends on your specific question and prior knowledge about the data. If you want "blind" dimensionality reduction, PCA is a good starting point. If you want some "informed" dimensionality reduction (e.g., you know that there is some circular structure that bears information relevant to your substantial question), have a look at kernel PCA. $\endgroup$ – Brandmaier Feb 13 '17 at 17:44
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You are looking for Multidimensional Scaling (MDS):

An MDS algorithm aims to place each object in N-dimensional space such that the between-object distances are preserved as well as possible. Each object is then assigned coordinates in each of the N dimensions. The number of dimensions of an MDS plot N can exceed 2 and is specified a priori. Choosing N=2 optimizes the object locations for a two-dimensional scatterplot.

There are many algorithms and libraries for many common software packages. This thread discusses MDS and PCA. Or you could browse through the tag.

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There are several ways you can project data into lower dimension. Few of them are:

  1. Perform Principal Component Analysis—take only the components where the variance is significant. It might not preserve the topology of the original space means points close to each other in the original space might not be close to each other in the reduced dimension space after PCA.

  2. Self organizing maps—This is another unsupervised method of training a grid of neurons in a lower dimensional lattice (preferably two dimension) so that each point maps to one of the neurons in the lower dimensional lattice. The good thing about self organizing maps is the topology of the original space is almost preserved in the lower dimensional space i.e points close to each other in the original space remains close to each other in the reduced lower dimensional lattice.

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