A Coupled Alpha Complex

TL;DR

This paper introduces a new homotopy equivalent alpha complex that maintains monotonicity, addressing computational inefficiencies of the standard alpha complex in topological data analysis.

Contribution

It presents a novel construction of a monotone alpha complex, with algorithms for its construction and homology computation, improving computational efficiency.

Findings

The new complex is homotopy equivalent to the alpha complex.

The size of the new complex is comparable to the standard alpha complex.

The construction preserves topological features while ensuring monotonicity.

Abstract

The alpha complex is a subset of the Delaunay triangulation and is often used in computational geometry and topology. One of the main drawbacks of using the alpha complex is that it is non-monotone, in the sense that if ${\cal X}\subset{\cal X}'$ it is not necessarily (and generically not) the case that the corresponding alpha complexes satisfy ${\cal A}_r({\cal X})\subset{\cal A}_r({\cal X}')$. The lack of monotonicity may introduce significant computational costs when using the alpha complex, and in some cases even render it unusable. In this work we present a new construction based on the alpha complex, that is homotopy equivalent to the alpha complex while maintaining monotonicity. We provide the formal definitions and algorithms required to construct this complex, and to compute its homology. In addition, we analyze the size of this complex in order to argue that it is not significantly more costly to use than the standard alpha complex.