Elastic Maps and Nets for Approximating Principal Manifolds and Their Application to Microarray Data Visualization

TL;DR

This paper introduces elastic maps, a non-linear method for constructing principal manifolds that improve data visualization and approximation, especially for complex datasets like microarray data, outperforming traditional PCA.

Contribution

The paper presents a new elastic maps approach for constructing principal manifolds with a simple quadratic smoothness penalty, enabling effective parallel implementation and application to microarray data visualization.

Findings

Elastic maps outperform linear PCA in data approximation.

Elastic maps better preserve local neighborhood structures.

The method effectively visualizes microarray data.

Abstract

Principal manifolds are defined as lines or surfaces passing through ``the middle'' of data distribution. Linear principal manifolds (Principal Components Analysis) are routinely used for dimension reduction, noise filtering and data visualization. Recently, methods for constructing non-linear principal manifolds were proposed, including our elastic maps approach which is based on a physical analogy with elastic membranes. We have developed a general geometric framework for constructing ``principal objects'' of various dimensions and topologies with the simplest quadratic form of the smoothness penalty which allows very effective parallel implementations. Our approach is implemented in three programming languages (C++, Java and Delphi) with two graphical user interfaces (VidaExpert http://bioinfo.curie.fr/projects/vidaexpert and ViMiDa http://bioinfo-out.curie.fr/projects/vimida applications). In this paper we overview the method of elastic maps and present in detail one of its major applications: the visualization of microarray data in bioinformatics. We show that the method of elastic maps outperforms linear PCA in terms of data approximation, representation of between-point distance structure, preservation of local point neighborhood and representing point classes in low-dimensional spaces.