Topological Grammars for Data Approximation

TL;DR

This paper introduces topological grammars for multidimensional data approximation, constructing low-dimensional principal complexes that generalize manifolds and optimize data fit through energy minimization.

Contribution

It proposes a novel framework using topological grammars to construct principal complexes that generalize manifolds for better data approximation.

Findings

Principal cubic complexes effectively approximate complex data topologies.

Energy minimization algorithms efficiently construct optimal complexes.

The method generalizes linear and nonlinear principal manifolds.

Abstract

A method of {\it topological grammars} is proposed for multidimensional data approximation. For data with complex topology we define a {\it principal cubic complex} of low dimension and given complexity that gives the best approximation for the dataset. This complex is a generalization of linear and non-linear principal manifolds and includes them as particular cases. The problem of optimal principal complex construction is transformed into a series of minimization problems for quadratic functionals. These quadratic functionals have a physically transparent interpretation in terms of elastic energy. For the energy computation, the whole complex is represented as a system of nodes and springs. Topologically, the principal complex is a product of one-dimensional continuums (represented by graphs), and the grammars describe how these continuums transform during the process of optimal complex construction. This factorization of the whole process onto one-dimensional transformations using minimization of quadratic energy functionals allow us to construct efficient algorithms.