Related papers: Computing the Matching Distance of 2-Parameter Per…
Connection matrices are a generalization of Morse boundary operators from the classical Morse theory for gradient vector fields. Developing an efficient computational framework for connection matrices is particularly important in the…
Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in…
We apply modern methods in computational topology to the task of discovering and characterizing phase transitions. As illustrations, we apply our method to four two-dimensional lattice spin models: the Ising, square ice, XY, and…
Dey and Xin (J.Appl.Comput.Top., 2022, arXiv:1904.03766) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators…
Distances on merge trees facilitate visual comparison of collections of scalar fields. Two desirable properties for these distances to exhibit are 1) the ability to discern between scalar fields which other, less complex topological…
Persistence diagrams, combining geometry and topology for an effective shape description used in pattern recognition, have already proven to be an effective tool for shape representation with respect to a certainfiltering function.…
Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in…
A persistence module with $m$ discrete parameters is a diagram of vector spaces indexed by the poset $\mathbb{N}^m$. If we are only interested in the large scale behavior of such a diagram, then we can consider two diagrams equivalent if…
Metric graphs are ubiquitous in science and engineering. For example, many data are drawn from hidden spaces that are graph-like, such as the cosmic web. A metric graph offers one of the simplest yet still meaningful ways to represent the…
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…
Calibration of expensive computer models with high-dimensional output fields can be approached via history matching. If the entire output field is matched, with patterns or correlations between locations or time points represented,…
In this work we study the interleaving distance between merge trees from a combinatorial point of view. We use a particular type of matching between trees to obtain a novel formulation of the distance. With such formulation, we tackle the…
We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse…
Multiparameter persistent homology has been largely neglected as an input to machine learning algorithms. We consider the use of lattice-based convolutional neural network layers as a tool for the analysis of features arising from…
Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring…
In this article we propose a novel approach for comparing the persistent homology representations of two spaces (filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes,…
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…
This work incorporates topological features via persistence diagrams to classify point cloud data arising from materials science. Persistence diagrams are multisets summarizing the connectedness and holes of given data. A new distance on…
Bisimulation metrics are powerful tools for measuring similarities between stochastic processes, and specifically Markov chains. Recent advances have uncovered that bisimulation metrics are, in fact, optimal-transport distances, which has…
A central aspect of robotic motion planning is collision avoidance, where a multitude of different approaches are currently in use. Optimization-based motion planning is one method, that often heavily relies on distance computations between…