Persistence weighted Gaussian kernel for topological data analysis

Abstract

Topological data analysis (TDA) is an emerging mathematical concept for characterizing shapes in complex data. In TDA, persistence diagrams are widely recognized as a useful descriptor of data, and can distinguish robust and noisy topological properties. This paper proposes a kernel method on persistence diagrams to develop a statistical framework in TDA. The proposed kernel satisfies the stability property and provides explicit control on the effect of persistence. Furthermore, the method allows a fast approximation technique. The method is applied into practical data on proteins and oxide glasses, and the results show the advantage of our method compared to other relevant methods on persistence diagrams.

26 pages, 13 figures

References in corpus (3)

• Persistent homology analysis of protein structure, flexibility and folding
• Learning from Distributions via Support Measure Machines
• Principal Component Analysis of Persistent Homology Rank Functions with case studies of Spatial Point Patterns, Sphere Packing and Colloids

Cited by in corpus (37)

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• PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures
• The persistence landscape and some of its properties
• giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration
• Topological Persistence Machine of Phase Transitions
• Simplicial degree in complex networks. Applications of Topological Data Analysis to Network Science
• Topological Feature Vectors for Chatter Detection in Turning Processes
• Protein folding analysis using features obtained by persistent homology
• Topological exploration of artificial neuronal network dynamics
• Approximating Continuous Functions on Persistence Diagrams Using Template Functions
• Non-parametric Kernel-Based Estimation of Probability Distributions for Precipitation Modeling
• The density of expected persistence diagrams and its kernel based estimation
• Leveraging the Mathematics of Shape for Solar Magnetic Eruption Prediction
• Wasserstein Stability for Persistence Diagrams
• Kernel method for persistence diagrams via kernel embedding and weight factor
• Topological Data Analysis (TDA) for Time Series
• Topological Attention for Time Series Forecasting
• MONK -- Outlier-Robust Mean Embedding Estimation by Median-of-Means
• Limit theorems for persistence diagrams
• A persistent homology approach to heart rate variability analysis with an application to sleep-wake classification
• Persistence Diagrams with Linear Machine Learning Models
• Adaptive Partitioning for Template Functions on Persistence Diagrams
• RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds
• Distribution Regression for Sequential Data
• Limit Theorems for the Sum of Persistence Barcodes
• Nonparametric Estimation of Probability Density Functions of Random Persistence Diagrams
• ATOL: Measure Vectorization for Automatic Topologically-Oriented Learning
• Time series analysis using persistent homology of distance matrix
• Using Persistent Homology Topological Features to Characterize Medical Images: Case Studies on Lung and Brain Cancers
• Persistent Homology and Graphs Representation Learning
• TopoTxR: A Topological Biomarker for Predicting Treatment Response in Breast Cancer
• On the estimation of persistence intensity functions and linear representations of persistence diagrams
• Sparse-TDA: Sparse Realization of Topological Data Analysis for Multi-Way Classification
• Nonparametric Topological Layers in Neural Networks