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