Data complexity measured by principal graphs

TL;DR

This paper introduces new data complexity measures using principal cubic complexes, which generalize principal manifolds, to evaluate the geometric, structural, and construction complexity of datasets in multidimensional spaces.

Contribution

The paper proposes a novel framework for measuring data complexity with principal cubic complexes, including three natural complexity types and software implementation.

Findings

Computed complexity measures for various datasets

Demonstrated accuracy-complexity trade-offs in plots

Provided software tools for complexity measurement

Abstract

How to measure the complexity of a finite set of vectors embedded in a multidimensional space? This is a non-trivial question which can be approached in many different ways. Here we suggest a set of data complexity measures using universal approximators, principal cubic complexes. Principal cubic complexes generalise the notion of principal manifolds for datasets with non-trivial topologies. The type of the principal cubic complex is determined by its dimension and a grammar of elementary graph transformations. The simplest grammar produces principal trees. We introduce three natural types of data complexity: 1) geometric (deviation of the data's approximator from some "idealized" configuration, such as deviation from harmonicity); 2) structural (how many elements of a principal graph are needed to approximate the data), and 3) construction complexity (how many applications of elementary graph transformations are needed to construct the principal object starting from the simplest one). We compute these measures for several simulated and real-life data distributions and show them in the "accuracy-complexity" plots, helping to optimize the accuracy/complexity ratio. We discuss various issues connected with measuring data complexity. Software for computing data complexity measures from principal cubic complexes is provided as well.