arXiv subjects

Alexander McCleary

Publications and source records attributed to Alexander McCleary.

4 recordsLinked to original sources

arXiv2025

Tensor Computation of Euler Characteristic Functions and Transforms

The weighted Euler characteristic transform (WECT) and Euler characteristic function (ECF) have proven to be useful tools in a variety of applications. However, current methods for computing these functions are either not optimized for GPU computation or do not scale to higher-dimensional settings. In this work, we present a tensor-based framework for computing such topological descriptors which is highly optimized for GPU architectures and works in full generality across simplicial and cubical complexes of arbitrary dimension. Experimentally, the framework demonstrates significant speedups over existing methods when computing the WECT and ECF across a variety of two- and three-dimensional datasets. Computation of these transforms is implemented in a publicly available Python package called pyECT.

cs.CG↗

arXiv2022

Galois Connections in Persistent Homology

We present a new language for persistent homology in terms of Galois connections. This language has two main advantages over traditional approaches. First, it simplifies and unifies central concepts such as interleavings and matchings. Second, it provides access to Rota's Galois connection theorem -- a powerful tool with many potential applications in applied topology. To illustrate this, we use Rota's Galois connection theorem to give a substantially easier proof of the bottleneck stability theorem. Finally, we use this language to establish relationships between various notions of multiparameter persistence diagrams.

math.AT↗

arXiv2020

Edit Distance and Persistence Diagrams Over Lattices

We build a functorial pipeline for persistent homology. The input to this pipeline is a filtered simplicial complex indexed by any finite metric lattice and the output is a persistence diagram defined as the M\"obius inversion of its birth-death function. We adapt the Reeb graph edit distance to each of our categories and prove that both functors in our pipeline are $1$-Lipschitz making our pipeline stable. Our constructions generalize the classical persistence diagram and, in this setting, the bottleneck distance is strongly equivalent to the edit distance.

math.AT↗

arXiv2017

Darboux Calculus

We introduce a formalism to analyze partially defined functions between ordered sets. We show that our construction provides a uniform and conceptual approach to all the main definitions encountered in elementary real analysis including Dedekind cuts, limits and continuity.

math.CA↗