Symmetry in Data Mining and Analysis: A Unifying View based on Hierarchy

TL;DR

This paper presents a unifying framework for data analysis based on hierarchical structures and symmetry, linking concepts from number theory, topology, and various data analysis paradigms to address complex, high-dimensional data sets.

Contribution

It introduces a hierarchical perspective on data symmetry, integrating traditional data analysis methods with new approaches using p-adic numbers and ultrametric topology.

Findings

Hierarchical structures encapsulate traditional data analysis paradigms.

Symmetry concepts extend to high-dimensional, heterogeneous data sets.

Connections to number theory, logic, and symbolic dynamics are established.

Abstract

Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational or otherwise empirical domain of interest. "Structure" has long been understood as symmetry which can take many forms with respect to any transformation, including point, translational, rotational, and many others. Beginning with the role of number theory in expressing data, we show how we can naturally proceed to hierarchical structures. We show how this both encapsulates traditional paradigms in data analysis, and also opens up new perspectives towards issues that are on the order of the day, including data mining of massive, high dimensional, heterogeneous data sets. Linkages with other fields are also discussed including computational logic and symbolic dynamics. The structures in data surveyed here are based on hierarchy, represented as p-adic numbers or an ultrametric topology.