Path representations in multiparameter persistent homology

2025/07/31 by Sun, Xudong, Corbet, René, Marr, Carsten
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.23762

Abstract

Multiparameter persistence module can capture more topological differences across data instances compared to using a single parameter, where the well-studied matching distance investigates the distance along a straight line in the multiparameter space that gives the biggest difference. We propose to generalize the straight line to a monotone path filtration and offer software implementations.

Citations

• ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery
• An Introduction to Multiparameter Persistence
• The Shift-Dimension of Multipersistence Modules
• Computing the Multicover Bifiltration
• Efficient Approximation of the Matching Distance for 2-parameter persistence
• Nonembeddability of Persistence Diagrams with p>2 Wasserstein Metric
• Reinforcement learning
• Topological Autoencoders
• Multiparameter Persistence Landscapes
• Exact computation of the matching distance on 2-parameter persistence modules
• Computing the interleaving distance is NP-hard
• Generalized persistence analysis based on stable rank invariant
• On the geometrical properties of the coherent matching distance in 2D persistent homology
• Data structures for real multiparameter persistence modules
• Stratifying multiparameter persistent homology
• Sliced Wasserstein Kernel for Persistence Diagrams
• Geometry Helps to Compare Persistence Diagrams
• Interactive Visualization of 2-D Persistence Modules
• A Stable Multi-Scale Kernel for Topological Machine Learning
• The rank invariant stability via interleavings
• Fréchet Means for Distributions of Persistence diagrams
• Fréchet Means for Distributions of Persistence Diagrams
• Probability measures on the space of persistence diagrams

Related

• Computing the Matching Distance of 2-Parameter Persistence Modules from Critical Values
• An Introduction to Multiparameter Persistence
• Persistent Homology with Path-Representable Distances on Graph Data
• Persistent Path Laplacian
• Bipath Persistence