I am trying to give some theoretical foundations to the intuitive idea that three branches of mathematics are indeed tightly connected, specifically: clustering, graph-theory and principal components.
I would like to post this as a very open question, without commenting too much on why I believe these fields are connected, so that everyone can freely elaborate according to their intuition or academic knowledge.
On a high-level, clustering is aimed at separating objects into groups by making the objects belonging to the same group as similar as possible to each other.
Graph-theory, instead, tries to topologically represent the relationships between several objects. As such, the relative location of an object in a space is dependent on its similarity to neighbour objects. Somewhere, I have even read: "Clustering: an application of Minimum Spanning Tree"!
Finally, the link between clustering and PCA has been well discussed here:
What is the relation between k-means clustering and PCA?
So my question is: is there any proven mathematical link between these fields? it would be great if someone could also share references to academic papers that explored these connections.
