Reconstructing linearly embedded graphs: A first step to stratified space learning

<p style='text-indent:20px;'>In this paper, we consider the simplest class of stratified spaces – linearly embedded graphs. We present an algorithm that learns the abstract structure of an embedded graph and models the specific embedding from a point cloud sampled from it. We use tools and inspiration from computational geometry, algebraic topology, and topological data analysis and prove the correctness of the identified abstract structure under assumptions on the embedding. The algorithm is implemented in the Julia package Skyler, which we used for the numerical simulations in this paper.</p>

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Inferring Quality in Point Cloud-based 3D Printed Objects using Topological Data Analysis

Computer-Aided Design and Applications ◽

10.14733/cadaps.2019.519-527 ◽

2018 ◽

Vol 16 (3) ◽

pp. 519-527

Author(s):

Paul Rosen ◽

Mustafa Hajij ◽

Junyi Tu ◽

Tanvirul Arafin ◽

Les Piegl

Keyword(s):

Data Analysis ◽

Point Cloud ◽

Topological Data Analysis ◽

3D Printed ◽

Topological Data

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Persistent homology approach distinguishes potential pattern between “Early” and “Not Early” hepatic decompensation groups using MRI modalities

Current Directions in Biomedical Engineering ◽

10.1515/cdbme-2021-2124 ◽

2021 ◽

Vol 7 (2) ◽

pp. 488-491

Author(s):

Yashbir Singh ◽

William Jons ◽

Gian Marco Conte ◽

Jaidip Jagtap ◽

Kuan Zhang ◽

...

Keyword(s):

Data Analysis ◽

Liver Failure ◽

Algebraic Topology ◽

Persistent Homology ◽

Topological Data Analysis ◽

Topological Representation ◽

Hepatic Decompensation ◽

Topological Data

Abstract Primary sclerosis cholangitis (PSC) predisposes individuals to liver failure, but it is challenging for radiologists examining radiologic images to predict which patients with PSC will ultimately develop liver failure. Motivated by algebraic topology, a topological data analysis - inspired framework was adopted in the study of the imaging pattern between the “Early Decompensation” and “Not Early” groups. The results demonstrate that the proposed methodology discriminates “Early Decompensation” and “Not Early” groups. Our study is the first attempt to provide a topological representation-based method into early hepatic decompensation and not early groups.

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A COVID-19 Drug Repurposing Strategy Through Quantitative Homological Similarities by using a Topological Data Analysis Based Formalism

10.20944/preprints202012.0281.v1 ◽

2020 ◽

Author(s):

Raul Pérez-Moraga ◽

Jaume Forés-Martos ◽

Beatriz Suay ◽

Jean-Louis Duval ◽

Antonio Falcó ◽

...

Keyword(s):

Data Analysis ◽

Algebraic Topology ◽

Protein Structures ◽

Three Dimensional ◽

Drug Repurposing ◽

Topological Data Analysis ◽

Drug Identification ◽

Global Pandemic ◽

Candidate Drug ◽

Topological Data

Since its emergence in March 2020, the SARS-CoV-2 global pandemic has produced more than 65 million cases and one point five million deaths worldwide. Despite the enormous efforts carried out by the scientific community, no effective treatments have been developed to date. We created a novel computational pipeline aimed to speed up the process of repurposable candidate drug identification. Compared with current drug repurposing methodologies, our strategy is centered on filtering the best candidate among all selected targets focused on the introduction of a mathematical formalism motivated by recent advances in the fields of algebraic topology and topological data analysis (TDA). This formalism allows us to compare three-dimensional protein structures. Its use in conjunction with two in silico validation strategies (molecular docking and transcriptomic analyses) allowed us to identify a set of potential drug repurposing candidates targeting three viral proteins (3CL viral protease, NSP15 endoribonuclease, and NSP12 RNA-dependent RNA polymerase), which included rutin, dexamethasone, and vemurafenib among others. To our knowledge, it is the first time that a TDA based strategy has been used to compare a massive amount of protein structures with the final objective of performing drug repurposing

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Using Topological Data Analysis to Infer the Quality in Point Cloud-based 3D Printed Objects

10.14733/cadconfp.2018.66-70 ◽

2018 ◽

Author(s):

Paul Rosen ◽

Mustafa Hajij ◽

Junyi Tu ◽

Tanvirul Arafin ◽

Les Piegl

Keyword(s):

Data Analysis ◽

Point Cloud ◽

Topological Data Analysis ◽

3D Printed ◽

Topological Data

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Exploring Equilibria in Stochastic Delay Differential Equations Using Persistent Homology

Volume 8: 26th Conference on Mechanical Vibration and Noise ◽

10.1115/detc2014-35655 ◽

2014 ◽

Cited By ~ 1

Author(s):

Firas A. Khasawneh ◽

Elizabeth Munch

Keyword(s):

Data Analysis ◽

Point Cloud ◽

Dimensional Space ◽

Persistent Homology ◽

Topological Data Analysis ◽

Stochastic Delay Differential Equations ◽

Machining Processes ◽

Automatic Data ◽

Stochastic Delay ◽

Delay Differential

This paper explores the possibility of using techniques from topological data analysis for studying datasets generated from dynamical systems described by stochastic delay equations. The dataset is generated using Euler-Maryuama simulation for two first order systems with stochastic parameters drawn from a normal distribution. The first system contains additive noise whereas the second one contains parametric or multiplicative noise. Using Taken’s embedding, the dataset is converted into a point cloud in a high-dimensional space. Persistent homology is then employed to analyze the structure of the point cloud in order to study equilibria and periodic solutions of the underlying system. Our results show that the persistent homology successfully differentiates between different types of equilibria. Therefore, we believe this approach will prove useful for automatic data analysis of vibration measurements. For example, our approach can be used in machining processes for chatter detection and prevention.

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Contributions to Persistence Theory

Annals of West University of Timisoara - Mathematics and Computer Science ◽

10.2478/awutm-2014-0012 ◽

2014 ◽

Vol 52 (2) ◽

Cited By ~ 1

Author(s):

Dong Du

Keyword(s):

Data Analysis ◽

Algebraic Topology ◽

Point Cloud ◽

Point Cloud Data ◽

Qualitative Information ◽

Cloud Data ◽

Bar Codes ◽

Qualitative Understanding ◽

Rips Complex ◽

Persistence Theory

AbstractPersistence theory discussed in this paper is an application of algebraic topology (Morse Theory [29]) to Data Analysis, precisely to qualitative understanding of point cloud data, or PCD for short. PCD can be geometrized as a filtration of simplicial complexes (Vietoris-Rips complex [25] [36]) and the homology changes of these complexes provide qualitative information about the data. Bar codes describe the changes in homology with coefficients in a fixed field. When the coefficient field is ℤ

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Application of Topological Data Analysis to Multi-Resolution Matching of Aerosol Optical Depth Maps

Frontiers in Environmental Science ◽

10.3389/fenvs.2021.684716 ◽

2021 ◽

Vol 9 ◽

Author(s):

Dorcas Ofori-Boateng ◽

Huikyo Lee ◽

Krzysztof M. Gorski ◽

Michael J. Garay ◽

Yulia R. Gel

Keyword(s):

Data Analysis ◽

Power Systems ◽

Aerosol Optical Depth ◽

Optical Depth ◽

Algebraic Topology ◽

Data Science ◽

Earth Science ◽

Study Data ◽

Topological Data Analysis ◽

Topological Data

Topological data analysis (TDA) combines concepts from algebraic topology, machine learning, statistics, and data science which allow us to study data in terms of their latent shape properties. Despite the use of TDA in a broad range of applications, from neuroscience to power systems to finance, the utility of TDA in Earth science applications is yet untapped. The current study aims to offer a new approach for analyzing multi-resolution Earth science datasets using the concept of data shape and associated intrinsic topological data characteristics. In particular, we develop a new topological approach to quantitatively compare two maps of geophysical variables at different spatial resolutions. We illustrate the proposed methodology by applying TDA to aerosol optical depth (AOD) datasets from the Goddard Earth Observing System, Version 5 (GEOS-5) model over the Middle East. Our results show that, contrary to the existing approaches, TDA allows for systematic and reliable comparison of spatial patterns from different observational and model datasets without regridding the datasets into common grids.

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