Related papers: Persistent Homology for Structural Characterizatio…
We develop a method for analyzing spatial and spatiotemporal anomalies in geospatial data using topological data analysis (TDA). To do this, we use persistent homology (PH), which allows one to algorithmically detect geometric voids in a…
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
In medical image analysis, feature engineering plays an important role in the design and performance of machine learning models. Persistent homology (PH), from the field of topological data analysis (TDA), demonstrates robustness and…
Persistent homology (PH) is a powerful mathematical method to automatically extract relevant insights from images, such as those obtained by high-resolution imaging devices like electron microscopes or new-generation telescopes. However,…
We apply persistent homology to the task of discovering and characterizing phase transitions, using lattice spin models from statistical physics for working examples. Persistence images provide a useful representation of the homological…
This article introduces an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent…
Persistent homology (PH) is an approach to topological data analysis (TDA) that computes multi-scale topologically invariant properties of high-dimensional data that are robust to noise. While PH has revealed useful patterns across various…
Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we…
Persistent homology (PH) is a popular tool for topological data analysis that has found applications across diverse areas of research. It provides a rigorous method to compute robust topological features in discrete experimental…
Let $K$ be a finite simplicial, cubical, delta or CW complex. The persistence map $\mathrm{PH}$ takes a filter $f:K \rightarrow \mathbb{R}$ as input and returns the barcodes $\mathrm{PH}(f)$ of the associated sublevel set persistent…
The training of neural networks is usually monitored with a validation (holdout) set to estimate the generalization of the model. This is done instead of measuring intrinsic properties of the model to determine whether it is learning…
Persistent homology (PH) is one of the most popular methods in Topological Data Analysis. Even though PH has been used in many different types of applications, the reasons behind its success remain elusive; in particular, it is not known…
Predicting material properties of disordered systems remains a long-standing and formidable challenge in rational materials design. To address this issue, we introduce an automated software framework capable of modeling partial occupation…
We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the…
Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales, then rendering a family of…
The field of mathematical morphology offers well-studied techniques for image processing. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis.…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the…
Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much…
Qualitative methods such as the linear sampling method and the factorization method reconstruct acoustic scatterers through sampling indicators. In practice, these indicators are gray-scale fields on a prescribed sampling window and a…