Persistence Images: A Stable Vector Representation of Persistent Homology

Abstract

Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.

Version 3 contains updated theoretical results supporting methodology; expanded discussion of related works; extended list of references; extended applications section; additional experimental results and new figures

References in corpus (7)

• Statistical topological data analysis using persistence landscapes
• Topological Data Analysis of Biological Aggregation Models
• Probabilistic Fréchet Means for Time Varying Persistence Diagrams
• Statistical Analysis of Persistence Intensity Functions
• Topological and Statistical Behavior Classifiers for Tracking Applications
• Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis
• Topological descriptors for 3D surface analysis

Cited by in corpus (21)

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• Statistical Analysis of Persistence Intensity Functions
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• Topological Feature Vectors for Chatter Detection in Turning Processes
• A fractal dimension for measures via persistent homology
• Noise robustness of persistent homology on greyscale images, across filtrations and signatures
• Machine learning and Topological data analysis identify unique features of human papillae in 3D scans
• Characterising epithelial tissues using persistent entropy
• Classification of Stochastic Processes with Topological Data Analysis
• Nonparametric Estimation of Probability Density Functions of Random Persistence Diagrams
• Time series analysis using persistent homology of distance matrix
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• TopOC: Topological Deep Learning for Ovarian and Breast Cancer Diagnosis
• Topology Estimation of Simulated 4D Image Data by Combining Downscaling and Convolutional Neural Networks
• Stability for Inference with Persistent Homology Rank Functions
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