Geometrical complexity of data approximators

TL;DR

This paper introduces a new measure of geometrical complexity for data approximators, enabling comparison across different approximation methods to balance accuracy and complexity.

Contribution

It proposes a universal geometrical complexity measure applicable to various data approximators, facilitating cross-method comparison.

Findings

The measure allows effective comparison of different data approximation methods.

It helps in balancing approximation accuracy with model complexity.

The approach is applicable to multiple types of data approximators.

Abstract

There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types of principal curves and principal trees, and so on. For each type of approximators the measure of the approximator complexity was developed too. These measures are necessary to find the balance between accuracy and complexity and to define the optimal approximations of a given type. We propose a measure of complexity (geometrical complexity) which is applicable to approximators of several types and which allows comparing data approximations of different types.