PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures (1904.09378v4)

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.

• The paper introduces PersLay, a novel layer that integrates persistence diagrams into neural networks for end-to-end learning.
• It proposes extended persistence diagrams using Heat Kernel Signatures to robustly capture the topological features of graph data.
• Experimental results on synthetic and real-world datasets demonstrate the method's efficiency and superior performance in graph classification.

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

The paper at hand addresses the challenging task of incorporating persistence diagrams, which are central components of Topological Data Analysis (TDA), into machine learning workflows, especially within neural network architectures. Persistence diagrams encode topological properties of data, such as connected components and loops, and have been applied across various domains, including signal analysis and shape recognition. Nevertheless, their direct deployment in machine learning remains complex due to the non-Hilbert nature of their metric space, complicating the representation and manipulation under standard learning methodologies.

Contributions and Methods

The paper presents two primary contributions that aim to simplify the integration of persistence diagrams into learning algorithms, particularly focusing on graph-structured data:

1. Introduction of Extended Persistence Diagrams Based on Heat Kernel Signatures (HKS): The authors propose a new family of topological signatures calculated from the Heat Kernel Signatures of graphs. These extended persistence diagrams are designed to enrich the topological summary of graphs by incorporating additional stable features. They inherently provide robustness with respect to the graph's structure and the associated parameters, which is validated through theoretical stability proofs. This framework is applied to a variety of datasets, where it demonstrates effectiveness, particularly in handling large-scale and complex graph data.
2. Development of PersLay, a Neural Network Layer: To handle the data representation issue, the authors propose a novel neural network layer named PersLay, which can transform persistence diagrams into suitable vectorizations. PersLay is versatile enough to encompass most known vectorization techniques for persistence diagrams by allowing specification of weight functions and transformation operations. It can seamlessly integrate into existing neural network architectures, thus enabling end-to-end learning wherein the vectorization of persistence diagrams is optimized with the task-specific learning objectives.

Experimental Validation

The paper provides comprehensive experimental validation across synthetic and real-world datasets:

• Synthetic Dynamical Systems Dataset:

On a synthetic dataset simulating orbits from different dynamical systems, this approach surpassed previous kernel-based methods in classification tasks, indicating its computational efficiency and scalability to larger datasets.

• Graph Classification Tasks:

The proposed method was benchmarked on various real-world graph datasets from social network analysis to bioinformatics. The results were competitive and often superior to existing state-of-the-art approaches. This demonstrates both the theoretical robustness and practical applicability of PersLay and the new graph signatures.

Implications and Future Directions

Practically, this work has the potential to improve machine learning applications in fields reliant on graph-structured data, such as chemistry, biology, and social sciences, by providing a more expressive topological feature extraction method. Theoretically, the combination of neural networks with topological methods paves the way for future research at the intersection of TDA and deep learning. Exploring more diverse network architectures and tasks, alongside further refinement of the feature extraction paradigms introduced here, could yield substantial advancements in model capabilities and understanding of complex data structures.

Overall, this paper contributes significantly to the TDA and machine learning communities by offering a robust, scalable, and theoretically grounded framework for processing topological insights embedded within data. The open availability of PersLay as part of established software libraries also ensures that the broader research community can leverage these advancements effectively.