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PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures
Mathieu Carrière, Frédéric Chazal, Yuichi Ike, Théo Lacombe, Martin Royer, Yuhei Umeda
TL;DR
Persistence diagrams are powerful topological descriptors but awkward inputs for machine learning because their space lacks Hilbert-space structure and existing vectorizations can be inflexible or costly. The paper builds stable graph descriptors from extended persistence and Heat Kernel Signatures, then introduces PersLay to learn task-specific vectorizations. The approach achieves state-of-the-art synthetic orbit-classification results and competitive real-graph classification results while scaling beyond persistence-diagram kernel methods.
Problem
Persistence diagrams are difficult machine-learning inputs, while existing vectorizations may have few trainable parameters and kernel methods can be expensive for large datasets or diagrams.
Method
The paper constructs extended persistence diagrams from graph Heat Kernel Signatures and introduces PersLay, a differentiable neural layer that learns persistence-diagram vectorizations.
Results
The approach achieves state-of-the-art results on synthetic orbit classification and competitive results on several real-life graph-classification problems, while working at larger scales than persistence-diagram kernel methods.
Takeaways & Limitations
PersLay provides a general task-wise framework that encompasses most common persistence-diagram vectorizations and supports topological descriptors in neural-network-based learning tasks.
Takeaways & Limitations
Optimizing the Heat Kernel Signature diffusion parameter requires recomputing all persistence diagrams at each training epoch, greatly increasing experiment runtime.
Abstract
from arXiv · showhide
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.
1 Introduction
Persistence diagrams capture topological information but are difficult machine-learning inputs because their space lacks key vector-space structure. The paper introduces stable graph signatures and a learnable, versatile neural vectorization framework to address these limitations.
- Motivation: Persistence diagrams encode topological features such as connected components, loops, and cavities through points in R2 whose coordinates represent feature size.Their topological information complements more classical data descriptors across applications including signal analysis, materials, cellular data, and shape recognition.
- Motivation: The space of persistence diagrams lacks a fixed number of points and basic operations such as addition and scalar multiplication, complicating machine-learning use.Existing approaches therefore embed diagrams into finite-dimensional Euclidean spaces or implicit infinite-dimensional Hilbert spaces through kernels.
- Open challenges: Most existing vectorizations have few trainable parameters, making task-specific selection difficult, while kernel methods become expensive in time and memory for large datasets or diagrams.Kernel matrix-entry computation is at least linear and often quadratic in the number of persistence-diagram points.
- Contributions: PersLay learns persistence-diagram vectorizations for the target task and can recover most common literature vectorizations through parameter choices.The framework uses neural networks to process persistence diagrams rather than relying solely on static vectorization methods.
- Contributions: The paper contributes extended persistence diagrams built from graph Heat Kernel Signatures and a differentiable PersLay layer that generalizes common persistence-diagram vectorizations.The graph signatures are designed to capture graph topology, while PersLay provides a general neural processing framework.
- Applications: The work combines these contributions for graph classification across benchmark datasets from biology, chemistry, and social sciences.The introduction frames this combination as an application to real-graph classification.
2 Extended persistence diagrams
The paper constructs graph topological signatures by applying extended persistence to Heat Kernel Signatures, and proves stability with respect to graph perturbations and the diffusion parameter.
- Extended persistence: Ordinary persistence can leave graph loops alive indefinitely, whereas extended persistence also analyzes superlevel sets to record their finite death times.This retains more topological information while keeping diagram coordinates finite.
- Extended persistence: An extended persistence diagram Dg(G, f) is formed by converting feature intervals [αb, αd] into a multiset of points in R2.For graphs, the resulting points represent four feature types: branches, loops, and connected components.
- Heat Kernel Signatures: The graph Heat Kernel Signature is a vertex function derived from the spectral decomposition of the normalized graph Laplacian.HKS provides the filtration function used to construct Dg(G, t).
- Heat Kernel Signatures: The HKS-based diagrams encode global topological structure in addition to the distributions of HKS values used by prior graph signatures.The paper denotes this construction by Dg(G, t) and establishes its stability in bottleneck distance.
- Stability: The HKS-based persistence diagrams are stable under sufficiently small graph perturbations and are 2-Lipschitz continuous in diffusion parameter t.The graph-perturbation bound uses a constant depending on t and the spectrum of the graph Laplacian.
- Stability: The map t ↦ Dg(G, t) is robust to the choice of diffusion parameter, with empirical illustrations reported in the appendix.This addresses parameter selection for statistical and learning applications.
3 Neural network learning with PersLay
PersLay learns permutation-invariant vectorizations of persistence diagrams through trainable point transformations, weights, and aggregation operations. Its general formulation recovers many existing vectorizations and supports large-scale classification experiments.
- PersLay: PersLay is a general neural-network layer that learns persistence-diagram vectorizations for the task being solved.Unlike static methods, its vectorization is optimized during training.
- PersLay: The layer applies a point transformation and weight to each diagram point, then combines the results with a permutation-invariant operation.The operation may be minimum, maximum, sum, or kth largest value.
- PersLay: The weight and point-transformation parameters can be optimized by back-propagation, and PersLay can be composed with any neural-network architecture ρ.This makes the vectorization trainable within broader neural architectures.
- Connections to prior methods: The formulation recovers most classical persistence-diagram vectorizations and treats the choice of point transformation as a hyperparameter.Examples include persistence landscapes, silhouettes, surfaces, and related kernel constructions.
- Stability: Continuity results characterize when sum-based learned vectorizations are continuous or stable, while continuity can constrain weights near the diagonal.The paper notes that such constraints may be undesirable when near-diagonal points matter for a learning task.
- Experiments: On ORBIT5K and ORBIT100K, the experiments report improved prior results and further increased classification accuracy as more observations become available.ORBIT100K contains 100,000 point clouds, a scale described as beyond kernel methods, and accuracy is averaged over 100 runs.
4 Application to graph classification
PersLay is evaluated on synthetic orbit and real-world graph classification datasets using a simple two-layer architecture that combines persistence-diagram features with graph features. Its performance is generally competitive, while topology appears less discriminative for the NCI datasets.
- Architecture: PersLay uses a two-layer network in which the first layer processes persistence diagrams and the final fully connected layer produces predictions.The architecture is intentionally simple to emphasize understanding rather than maximum performance.
- Datasets: The experiments cover social, medical, and biological graph datasets, including REDDIT5K, REDDIT12K, COLLAB, IMDB-B, IMDB-M, COX2, DHFR, MUTAG, PROTEINS, NCI1, and NCI109.These datasets are used as benchmarks for graph classification.
- Architecture: Each graph is represented by a set of persistence diagrams processed by independent PersLay instances using trainable w and φ functions and a fixed permutation-invariant operator.The graph-classification architecture is illustrated in Figure 3.
- Results: On REDDIT5K and REDDIT12K, a prior persistence-diagram network reported 54.5% and 44.5% accuracy, respectively.These results provide an additional comparison for persistence-diagram-based graph classification.
- Features: Topological features are combined with normalized graph-Laplacian eigenvalues and HKS deciles before classification.The impact of the topological features is assessed through an ablation study.
- Evaluation: 10 ten-fold evaluations report both average and best ten-fold classification accuracy, with the mean intended as the more robust evaluation.The best single ten-fold result is reported for comparison with competing methods.
- Results: In most cases, PersLay is comparable with state-of-the-art results, but topology-based methods have mediocre performance on the NCI datasets.The authors suggest that topology is not discriminative for these datasets.
5 Conclusion
The paper introduces stable graph topological signatures and PersLay, a task-wise trainable layer that generalizes common persistence-diagram vectorizations. Experiments report state-of-the-art synthetic orbit classification and competitive real-world graph classification at larger scales than persistence-diagram kernels.
- Contributions: The paper introduces stable, well-formed topological signatures for graphs and PersLay, a versatile neural-network layer for persistence diagrams.PersLay generalizes many existing vectorization techniques while optimizing them for the task.
- Contributions: PersLay generalizes most literature vectorizations while learning their parameters task-wise.This makes the framework adaptable to the learning task being solved.
- Results: The approach achieves state-of-the-art results on synthetic orbit classification and competitive results on several real-life graph classification problems.The conclusion summarizes performance across both synthetic and real-world settings.
- Results: The experiments operate at larger scales than persistence-diagram kernel methods and use a simpler architecture than most neural-network competitors.The paper presents these properties as part of the demonstrated efficiency of the approach.
- Reproducibility: The implementation is publicly available through the PersLay GitHub repository and the Gudhi4 library.The code is described as freely available.
A Proofs of stability theorems
The stability proofs combine perturbation bounds for graph Laplacians and heat-kernel signatures with persistence-diagram stability under the bottleneck distance. The arguments cover graph perturbations and changes in the diffusion parameter.
- Diagram distances: Persistence diagrams are represented as multisets that may have different numbers of points, with diagonal matching enabling bijections between diagrams.The diagonal Δ is included in the matching construction used to define diagram distances.
- Proof strategy: Theorem 2.2 follows by combining a graph-Laplacian perturbation theorem with general stability results for persistence diagrams.The perturbation theorem controls changes in heat-kernel signatures, while persistence stability transfers these bounds to diagrams.
- Graph perturbations: The Laplacian perturbation result assumes two graphs have the same number of vertices and sufficiently small Frobenius-norm perturbation of the Laplacian.When eigenvalues are distinct, the bound depends on their smallest separation δ and separates the cases of fully distinct and repeated eigenvalues.
- Extended-persistence subtypes: The resulting stability statements also apply to the extended-persistence subtypes Ord0, Rel1, and Ext+.This extends the stability guarantee beyond a single diagram subtype.
- Diffusion-parameter stability: For a fixed graph, the heat-kernel signature varies continuously with the diffusion parameter because each spectral factor exp(−tλ_k) is 2-Lipschitz for t > 0.The proof bounds the pointwise signature difference for two positive diffusion parameters and then invokes bottleneck-distance stability.
B Datasets description
The paper summarizes key information for each dataset used in both experiments and illustrates the generated orbits in Figure 4.
- Tables 3 and 4 summarize key information for each dataset used in both experiments.
- Figure 4 illustrates the orbits generated in Section 3.2.
- The dataset summaries and orbit illustration provide experiment-related information.
C.1 Weight learning
The learned weight grid identifies persistence-diagram regions that are discriminative for classification, illustrating task-adaptive weighting.
- Weight learning: Training transforms the 20 × 20 weight grid so activated cells highlight persistence-diagram regions that are discriminative for MUTAG classification.The grid is shown before and after training, with separate extended-diagram types for branches, connected components, and loops.
C.2 Selection of HKS diffusion parameter
The diffusion parameter t can be learned, but on MUTAG it quickly converges and offers little practical benefit over fixing a log-scale value. Updating it also increases runtime because diagrams must be recomputed each epoch.
- Parameter sensitivity: The four persistence diagrams built from a MUTAG graph illustrate that Dg(G, t) changes smoothly with the HKS diffusion parameter.This visualizes the 2-Lipschitz stability result for fixed graphs.
- Parameter optimization: Across 40 MUTAG training epochs, t converges quickly from initializations 0.1, 1, and 10, with t0 = 10.0 remaining almost constant.The near-constant trajectory at t0 = 10.0 suggests that this initialization is locally optimal in the experiment.
- Practical trade-off: Learning t is not worthwhile in this experiment because the diagrams are not particularly sensitive to its choice.Each update requires recomputing diagrams for every training graph, significantly increasing running time.
C.3 Experimental settings
The experiments use specified persistence-diagram constructions, PersLay configurations, optimization settings, and dataset-specific evaluation protocols. These settings include HKS parameter variation and repeated accuracy measurements.
- Table 5 records the experimental settings used to generate the reported results, including diagrams, neural-network architectures, and optimizers.
- Alpha_d denotes persistence diagrams from Gudhi’s d-dimensional AlphaComplex filtration, while hks_t denotes HKS-based extended persistence diagrams with parameter t.
- prom(k) preprocesses diagrams by selecting the k points farthest from the diagonal.
- Figure 7 tracks the evolution of t 7→Dg(G, t) for one MUTAG graph over t ∈ [0.1, 100] on a logarithmic scale.
- Table 6 identifies the hyper-parameter settings used for the influence analysis and ablations, with single-parameter changes evaluated against fixed supplementary settings.
- PersLay channels use either Gaussian-grid point transformations with convolution or line transformations followed by permutation-equivariant functions.
- The ADAM configuration specifies learning rate λ, Exponential Moving Average decay d, and training duration e epochs.
C.4 Hyper-parameters influence
The analyses examine how additional spectral features, persistence diagrams, weight-function grid size, and the HKS parameter affect classification accuracy and runtime. Larger grids can overfit, while HKS-parameter sensitivity differs between datasets.
- The ablation study compares test accuracy from additional spectral features alone with accuracy from extended and ordinary persistence diagrams alone.
- Increasing the weight-function grid size improves training accuracy but can overfit at excessively large values.
- Grid-size increases have little effect on running time, whereas removing the grid substantially lowers runtime.
- On MUTAG, classification accuracy does not depend on the HKS parameter t across the tested range.
- The HKS-parameter experiment uses a 10-fold evaluation with 100 epochs and PersLay configuration Im(20,(),20).
- Table 7 provides complementary experimental results, while omitted standard deviations are noted for presentation clarity.