Michael Lesnick
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View article: Sparse Approximation of the Subdivision-Rips Bifiltration for Doubling Metrics
Sparse Approximation of the Subdivision-Rips Bifiltration for Doubling Metrics Open
The Vietoris-Rips filtration, the standard filtration on metric data in topological data analysis, is notoriously sensitive to outliers. Sheehy's subdivision-Rips bifiltration $\mathcal{SR}(-)$ is a density-sensitive refinement that is rob…
View article: Nerve Models of Subdivision Bifiltrations
Nerve Models of Subdivision Bifiltrations Open
We study the size of Sheehy's subdivision bifiltrations, up to homotopy. We focus in particular on the subdivision-Rips bifiltration $\mathcal{SR}(X)$ of a metric space $X$, the only density-sensitive bifiltration on metric spaces known to…
View article: Delaunay Bifiltrations of Functions on Point Clouds
Delaunay Bifiltrations of Functions on Point Clouds Open
The Delaunay filtration $\mathcal{D}_{\bullet}(X)$ of a point cloud $X\subset \mathbb{R}^d$ is a central tool of computational topology. Its use is justified by the topological equivalence of $\mathcal{D}_{\bullet}(X)$ and the offset (i.e.…
View article: Efficient two-parameter persistence computation via cohomology
Efficient two-parameter persistence computation via cohomology Open
Clearing is a simple but effective optimization for the standard algorithm of persistent homology (PH), which dramatically improves the speed and scalability of PH computations for Vietoris--Rips filtrations. Due to the quick growth of the…
View article: Efficient Two-Parameter Persistence Computation via Cohomology
Efficient Two-Parameter Persistence Computation via Cohomology Open
Clearing is a simple but effective optimization for the standard algorithm of persistent homology (ph), which dramatically improves the speed and scalability of ph computations for Vietoris-Rips filtrations. Due to the quick growth of the …
View article: An Introduction to Multiparameter Persistence
An Introduction to Multiparameter Persistence Open
In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately ca…
View article: The Universal 𝓁^p-Metric on Merge Trees
The Universal 𝓁^p-Metric on Merge Trees Open
Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an 𝓁^p-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds th…
View article: $\ell^p$-Distances on Multiparameter Persistence Modules
$\ell^p$-Distances on Multiparameter Persistence Modules Open
Motivated both by theoretical and practical considerations in topological data analysis, we generalize the $p$-Wasserstein distance on barcodes to multiparameter persistence modules. For each $p\in [1,\infty]$, we in fact introduce two suc…
View article: Computing the Multicover Bifiltration
Computing the Multicover Bifiltration Open
Given a finite set $A\subset\mathbb{R}^d$, let Cov$_{r,k}$ denote the set of all points within distance $r$ to at least $k$ points of $A$. Allowing $r$ and $k$ to vary, we obtain a 2-parameter family of spaces that grow larger when $r$ inc…
View article: Computing the Multicover Bifiltration
Computing the Multicover Bifiltration Open
Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called…
View article: Exact computation of the matching distance on 2-parameter persistence modules
Exact computation of the matching distance on 2-parameter persistence modules Open
The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonabl…
View article: Stability of 2-Parameter Persistent Homology
Stability of 2-Parameter Persistent Homology Open
The Čech and Rips constructions of persistent homology are stable with respect to perturbations of the input data. However, neither is robust to outliers, and both can be insensitive to topological structure of high-density regions of the …
View article: A Formal Framework for Cognitive Models of Multitasking
A Formal Framework for Cognitive Models of Multitasking Open
This note introduces mathematical foundations for modeling of human multitask performance. Using basic definitions from set theory and graph theory, we introduce formal definitions of the environment in which multitasks are performed, of a…
View article: Quantifying Genetic Innovation: Mathematical Foundations for the Topological Study of Reticulate Evolution
Quantifying Genetic Innovation: Mathematical Foundations for the Topological Study of Reticulate Evolution Open
A topological approach to the study of genetic recombination, based on persistent homology, was introduced by Chan, Carlsson, and Rabadán in 2013. This associates a sequence of signatures called barcodes to genomic data sampled from an evo…
View article: Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology
Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology Open
Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of fr…
View article: Exact Computation of the Matching Distance on 2-Parameter Persistence Modules
Exact Computation of the Matching Distance on 2-Parameter Persistence Modules Open
The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonabl…
View article: Feasibility of topological data analysis for event-related fMRI
Feasibility of topological data analysis for event-related fMRI Open
Recent fMRI research shows that perceptual and cognitive representations are instantiated in high-dimensional multivoxel patterns in the brain. However, the methods for detecting these representations are limited. Topological data analysis…
View article: Algebraic stability of zigzag persistence modules
Algebraic stability of zigzag persistence modules Open
The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of [math] –valued functions, the result was later cast in a more…
View article: Persistent Homology for Virtual Screening
Persistent Homology for Virtual Screening Open
Finding new medicines is one of the most important tasks of pharmaceutical companies. One of the best approaches to finding a new drug starts with answering this simple question: Given a known effective drug X, what are the top 100 molecul…
View article: PHoS: Persistent Homology for Virtual Screening
PHoS: Persistent Homology for Virtual Screening Open
Finding new medicines is one of the most important tasks of pharma- ceutical companies. One of the best approaches to finding a new drug starts with answering this simple question: Given a known effective drug X, what are the top 100 molec…
View article: Persistent Homology for Virtual Screening
Persistent Homology for Virtual Screening Open
Finding new medicines is one of the most important tasks of pharmaceutical companies. One of the best approaches to finding a new drug starts with answering this simple question: Given a known effective drug X, what are the top 100 mole…
View article: PHoS: Persistent Homology for Virtual Screening
PHoS: Persistent Homology for Virtual Screening Open
Finding new medicines is one of the most important tasks of pharma- ceutical companies. One of the best approaches to finding a new drug starts with answering this simple question: Given a known effective drug X, what are the top 100 molec…
View article: Universality of the Homotopy Interleaving Distance
Universality of the Homotopy Interleaving Distance Open
As a step towards establishing homotopy-theoretic foundations for topological data analysis (TDA), we introduce and study homotopy interleavings between filtered topological spaces. These are homotopy-invariant analogues of interleavings, …
View article: Interactive Visualization of 2-D Persistence Modules
Interactive Visualization of 2-D Persistence Modules Open
The goal of this work is to extend the standard persistent homology pipeline for exploratory data analysis to the 2-D persistence setting, in a practical, computationally efficient way. To this end, we introduce RIVET, a software tool for …