Natural methods of unsupervised topological alignment

TL;DR

This paper analyzes and generalizes methods for unsupervised topological alignment, focusing on coupled approaches that handle diverse data types, and explores mathematical foundations and potential applications involving advanced algebraic structures.

Contribution

It introduces harmonic generalizations of graph Laplacian and kernel methods for aligning heterogeneous data sets, expanding theoretical understanding and application scope.

Findings

Harmonious generalizations of graph Laplacian and kernel methods

Framework for coupling data sets of different natures

Discussion of applications involving hypercomplex numbers and Clifford algebras

Abstract

In the paper, we represent a comparison analysis of the methods of the topological alignment and extract the main mathematical principles forming the base of the concept. The main narrative is devoted to the so-called coupled methods dealing with the data sets of various nature. As a main theoretical result, we obtain harmonious generalizations of the graph Laplacian and kernel based methods with the central idea to find a natural structure coupling data sets of various nature. Finally, we discuss prospective applications and consider far reaching generalizations related to the hypercomplex numbers and Clifford algebras.