Principal Graphs and Manifolds

TL;DR

This paper introduces principal graphs and manifolds as advanced methods for approximating complex multidimensional data, generalizing principal components and k-means, using expectation/maximisation algorithms and graph grammar techniques.

Contribution

It presents a unifying framework for constructing principal graphs and manifolds, extending classical methods with new algorithms and complexity control mechanisms.

Findings

Principal graphs generalize principal components.

Expectation/maximisation algorithms are adapted for graph construction.

Graph grammar approach enables controlled complexity.

Abstract

In many physical, statistical, biological and other investigations it is desirable to approximate a system of points by objects of lower dimension and/or complexity. For this purpose, Karl Pearson invented principal component analysis in 1901 and found 'lines and planes of closest fit to system of points'. The famous k-means algorithm solves the approximation problem too, but by finite sets instead of lines and planes. This chapter gives a brief practical introduction into the methods of construction of general principal objects, i.e. objects embedded in the 'middle' of the multidimensional data set. As a basis, the unifying framework of mean squared distance approximation of finite datasets is selected. Principal graphs and manifolds are constructed as generalisations of principal components and k-means principal points. For this purpose, the family of expectation/maximisation algorithms with nearest generalisations is presented. Construction of principal graphs with controlled complexity is based on the graph grammar approach.