Principal manifolds and graphs in practice: from molecular biology to dynamical systems

TL;DR

This paper demonstrates the use of elastic principal graphs for non-linear data modeling across diverse fields, showing advantages over linear methods and proposing criteria for comparing these mappings.

Contribution

It introduces elastic principal graphs as a flexible non-linear modeling tool and provides practical applications and comparison criteria across multiple disciplines.

Findings

Non-linear models outperform linear ones in data approximation.

Elastic principal graphs effectively capture complex data structures.

Proposed criteria facilitate comparison between linear and non-linear mappings.

Abstract

We present several applications of non-linear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen's self-organizing maps, a class of artificial neural networks. On several examples we show advantages of using non-linear objects for data approximation in comparison to the linear ones. We propose four numerical criteria for comparing linear and non-linear mappings of datasets into the spaces of lower dimension. The examples are taken from comparative political science, from analysis of high-throughput data in molecular biology, from analysis of dynamical systems.