arXiv subjects

Alexander Rolle

Publications and source records attributed to Alexander Rolle.

7 recordsLinked to original sources

arXiv2022

The Degree-Rips Complexes of an Annulus with Outliers

The degree-Rips bifiltration is the most computable of the parameter-free, density-sensitive bifiltrations in topological data analysis. It is known that this construction is stable to small perturbations of the input data, but its robustness to outliers is not well understood. In recent work, Blumberg-Lesnick prove a result in this direction using the Prokhorov distance and homotopy interleavings. Based on experimental evaluation, they argue that a more refined approach is desirable, and suggest the framework of homology inference. Motivated by these experiments, we consider a probability measure that is uniform with high density on an annulus, and uniform with low density on the disc inside the annulus. We compute the degree-Rips complexes of this probability space up to homotopy type, using the Adamaszek-Adams computation of the Vietoris-Rips complexes of the circle. These degree-Rips complexes are the limit objects for the Blumberg-Lesnick experiments. We argue that the homology inference approach has strong explanatory power in this case, and suggest studying the limit objects directly as a strategy for further work.

math.AT↗

arXiv2022

A Unified View on the Functorial Nerve Theorem and its Variations

The nerve theorem is a basic result of algebraic topology that plays a central role in computational and applied aspects of the subject. In topological data analysis, one often needs a nerve theorem that is functorial in an appropriate sense, and furthermore one often needs a nerve theorem for closed covers as well as for open covers. While the techniques for proving such functorial nerve theorems have long been available, there is unfortunately no general-purpose, explicit treatment of this topic in the literature. We address this by proving a variety of functorial nerve theorems. First, we show how one can use elementary techniques to prove nerve theorems for covers by closed convex sets in Euclidean space, and for covers of a simplicial complex by subcomplexes. Then, we establish a more general, "unified" nerve theorem that subsumes many of the variants, using standard techniques from abstract homotopy theory.

math.AT↗

arXiv2021

Compression for 2-Parameter Persistent Homology

Compression aims to reduce the size of an input, while maintaining its relevant properties. For multi-parameter persistent homology, compression is a necessary step in any computational pipeline, since standard constructions lead to large inputs, and computational tasks in this area tend to be expensive. We propose two compression methods for chain complexes of free 2-parameter persistence modules. The first method extends the multi-chunk algorithm for one-parameter persistent homology, returning the smallest chain complex among all the ones quasi-isomorphic to the input. The second method produces minimal presentations of the homology of the input; it is based on an algorithm of Lesnick and Wright, but incorporates several improvements that lead to substantial performance gains. The two methods are complementary, and can be combined to compute minimal presentations for complexes with millions of generators in a few seconds. The methods have been implemented, and the software is publicly available. We report on experimental evaluations, which demonstrate substantial improvements in performance compared to previously available compression strategies.

math.AT↗

arXiv2020

Fast Minimal Presentations of Bi-graded Persistence Modules

Multi-parameter persistent homology is a recent branch of topological data analysis. In this area, data sets are investigated through the lens of homology with respect to two or more scale parameters. The high computational cost of many algorithms calls for a preprocessing step to reduce the input size. In general, a minimal presentation is the smallest possible representation of a persistence module. Lesnick and Wright proposed recently an algorithm (the LW-algorithm) for computing minimal presentations based on matrix reduction. In this work, we propose, implement and benchmark several improvements over the LW-algorithm. Most notably, we propose the use of priority queues to avoid extensive scanning of the matrix columns, which constitutes the computational bottleneck in the LW-algorithm, and we combine their algorithm with ideas from the multi-parameter chunk algorithm by Fugacci and Kerber. Our extensive experiments show that our algorithm outperforms the LW-algorithm and computes the minimal presentation for data sets with millions of simplices within a few seconds. Our software is publicly available.

math.AT↗

arXiv2020

Stable and consistent density-based clustering via multiparameter persistence

We consider the degree-Rips construction from topological data analysis, which provides a density-sensitive, multiparameter hierarchical clustering algorithm. We analyze its stability to perturbations of the input data using the correspondence-interleaving distance, a metric for hierarchical clusterings that we introduce. Taking certain one-parameter slices of degree-Rips recovers well-known methods for density-based clustering, but we show that these methods are unstable. However, we prove that degree-Rips, as a multiparameter object, is stable, and we propose an alternative approach for taking slices of degree-Rips, which yields a one-parameter hierarchical clustering algorithm with better stability properties. We prove that this algorithm is consistent, using the correspondence-interleaving distance. We provide an algorithm for extracting a single clustering from one-parameter hierarchical clusterings, which is stable with respect to the correspondence-interleaving distance. And, we integrate these methods into a pipeline for density-based clustering, which we call Persistable. Adapting tools from multiparameter persistent homology, we propose visualization tools that guide the selection of all parameters of the pipeline. We demonstrate Persistable on benchmark data sets, showing that it identifies multi-scale cluster structure in data.

math.ST↗

arXiv2019

Visualization tools for parameter selection in cluster analysis

We propose an algorithm, HPREF (Hierarchical Partitioning by Repeated Features), that produces a hierarchical partition of a set of clusterings of a fixed dataset, such as sets of clusterings produced by running a clustering algorithm with a range of parameters. This gives geometric structure to such sets of clustering, and can be used to visualize the set of results one obtains by running a clustering algorithm with a range of parameters.

stat.ML↗

arXiv2018

Central extensions and the classifying spaces of projective linear groups

If $G$ is a presheaf of groupoids on a small site, and $A$ is a sheaf of abelian groups, we prove that the sheaf cohomology group $H^2 (BG, A)$ is in bijection with a set of central extensions of $G$ by $A$. We use this result to study the motivic cohomology of the Nisnevich classifying space $BG$, when $G$ is a presheaf of groups on the smooth Nisnevich site over a field, and particularly when $G = PGL_{n}$. Finally, we show that, when $p$ is an odd prime, the Chow ring of the classifying space of $PGL_{p}$ injects into the motivic cohomology of the Nisnevich classifying space $BPGL_{p}$, over any field of characteristic zero containing a primitive $p^{th}$ root of unity.

math.AG↗