Persistent Homology for Random Fields and Complexes

Abstract

We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the algebraic topology of the excursion sets of random fields.

References in corpus (3)

• Random geometric complexes
• Euler Integration of Gaussian Random Fields and Persistent Homology
• Euler integration over definable functions

Cited by in corpus (29)

• A roadmap for the computation of persistent homology
• Topology and Geometry of Gaussian random fields I: on Betti Numbers, Euler characteristic and Minkowski functionals
• The persistence landscape and some of its properties
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• A Look into Chaos Detection through Topological Data Analysis
• Vortex knots in tangled quantum eigenfunctions
• Topological Data Analysis for Multivariate Time Series Data
• A fractal dimension for measures via persistent homology
• Finding self-similar behavior in quantum many-body dynamics via persistent homology
• Topological Learning for Brain Networks
• Topological signatures of interstellar magnetic fields - I. Betti numbers and persistence diagrams
• Topology of Musical Data
• Clustering of Local Extrema in Planck CMB maps
• Topological Data Analysis of Human Brain Networks Through Order Statistics
• Shape Theory. III. Comparative Theory of Backgound Independence
• Models of Random Knots
• Topological data analysis and diagnostics of compressible MHD turbulence
• Numerical analysis of topological characteristics of three-dimensional geological models of oil and gas fields
• Crackle: The Persistent Homology of Noise
• Delay Parameter Selection in Permutation Entropy Using Topological Data Analysis
• Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces
• Nonparametric Estimation of Probability Density Functions of Random Persistence Diagrams
• Impact of Redshift Space Distortion on Persistent Homology of cosmic matter density field
• Cosmic Strings-induced CMB anisotropies in light of Weighted Morphology
• Convergence of persistence diagram in the sparse regime
• Effects of Correlated Noise on the Performance of Persistence Based Dynamic State Detection Methods
• Understanding of a brain spatial map based on threshold-free function dendrogramization
• Random Colorings in Manifolds