PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures

Abstract

Persistence diagrams, the most common descriptors of Topological Data Analysis, encode topological properties of data and have already proved pivotal in many different applications of data science. However, since the (metric) space of persistence diagrams is not Hilbert, they end up being difficult inputs for most Machine Learning techniques. To address this concern, several vectorization methods have been put forward that embed persistence diagrams into either finite-dimensional Euclidean space or (implicit) infinite dimensional Hilbert space with kernels. In this work, we focus on persistence diagrams built on top of graphs. Relying on extended persistence theory and the so-called heat kernel signature, we show how graphs can be encoded by (extended) persistence diagrams in a provably stable way. We then propose a general and versatile framework for learning vectorizations of persistence diagrams, which encompasses most of the vectorization techniques used in the literature. We finally showcase the experimental strength of our setup by achieving competitive scores on classification tasks on real-life graph datasets.

References in corpus (9)

• Statistical topological data analysis using persistence landscapes
• NetLSD: Hearing the Shape of a Graph
• Persistence weighted Gaussian kernel for topological data analysis
• Sliced Wasserstein Kernel for Persistence Diagrams
• Learning metrics for persistence-based summaries and applications for graph classification
• Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams
• A Topology Layer for Machine Learning
• RetGK: Graph Kernels based on Return Probabilities of Random Walks
• Understanding the Topology and the Geometry of the Space of Persistence Diagrams via Optimal Partial Transport

Cited by in corpus (21)

• Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization
• A Survey of Vectorization Methods in Topological Data Analysis
• Learning metrics for persistence-based summaries and applications for graph classification
• Leveraging the Mathematics of Shape for Solar Magnetic Eruption Prediction
• A Topological Machine Learning Pipeline for Classification
• Noise robustness of persistent homology on greyscale images, across filtrations and signatures
• Cosmology with Persistent Homology: a Fisher Forecast
• Activation Landscapes as a Topological Summary of Neural Network Performance
• Elder-Rule-Staircodes for Augmented Metric Spaces
• Optimal quantization of the mean measure and applications to statistical learning
• Euler Characteristic Tools For Topological Data Analysis
• Understanding the Power of Persistence Pairing via Permutation Test
• Combining Geometric and Topological Information for Boundary Estimation
• ATOL: Measure Vectorization for Automatic Topologically-Oriented Learning
• The mergegram of a dendrogram and its stability
• Learning Hyperbolic Representations of Topological Features
• A Fast and Robust Method for Global Topological Functional Optimization
• Cosmology with Persistent Homology: Parameter Inference via Machine Learning
• Estimation and Quantization of Expected Persistence Diagrams
• Accelerating Iterated Persistent Homology Computations with Warm Starts
• Pull-back Geometry of Persistent Homology Encodings