[Paper Review] PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures

Mathieu Carrière, Frédéric Chazal+4 more|arXiv (Cornell University)|Apr 20, 2019

Topological and Geometric Data Analysis41 references75 citations

TL;DR

PersLay introduces a differentiable neural network layer to learn vectorizations of persistence diagrams derived from graphs, enabling end-to-end graph classification with competitive results and scalable performance.

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.

Motivation & Objective

• Introduce extended persistence diagrams built from Heat Kernel Signatures (HKS) on graphs and prove their stability.
• Develop PersLay, a neural network layer that learns vectorizations of persistence diagrams within a permutation-invariant framework.
• Show that PersLay generalizes common diagram vectorizations and is differentiable for end-to-end learning.
• Demonstrate scalability and competitive performance on synthetic dynamical-system datasets and real-world graph datasets.

Proposed method

• Define PersLay as a permutation-invariant layer: PersLay(Dg) = op({w(p) * phi(p) for p in Dg}).
• Provide three point transformation functions phi: triangle (Lambda), Gaussian (Gamma), and line (L) to map diagram points to feature vectors.
• Relate PersLay to existing vectorizations (landscapes, silhouettes, surfaces, kernels) as special cases.
• Allow trainable weight function w and representation phi, enabling continuous learning through backpropagation.
• Demonstrate that persistence diagrams used with PersLay can be stable under diagram distance dB and parameter t (diffusion parameter).
• Integrate PersLay into simple graph classifiers (two-layer network) and evaluate on standard graph benchmarks.

Experimental results

Research questions

• RQ1 How can persistence diagrams derived from graphs be incorporated into trainable neural networks effectively?
• RQ2 Can a single differentiable layer (PersLay) generalize and recover existing vectorizations of persistence diagrams?
• RQ3 Are extended persistence diagrams built from Heat Kernel Signatures stable and effective for graph classification tasks?
• RQ4 How does PersLay perform relative to kernel methods and standard graph neural networks on large-scale graph datasets?

Key findings

• PersLay achieves competitive accuracy on a range of graph classification benchmarks, often matching state-of-the-art methods.
• On the ORBIT5K and ORBIT100K dynamical-system datasets, PersLay outperforms several kernel-based methods and scales to very large datasets (ORBIT100K).
• The framework encompasses common vectorizations as special cases, enabling flexible learning of topological features for graphs.
• Extended persistence diagrams based on Heat Kernel Signatures are stable with respect to graph perturbations and the diffusion parameter t.
• A simple two-layer architecture using PersLay can achieve strong results without requiring complex graph neural networks.