EMP: Effective Multidimensional Persistence for Graph Representation Learning

Ignacio Segovia-Dominguez, Baris Coskunuzer, Cuneyt Gurcan Akcora, Yuzhou Chen, Zhiwei Zhen, Murat Kantarcioglu, Yulia Gel

Keywords: multiparameter persistence, topological data analysis, machine learning

Abstract:

Topological data analysis (TDA) has become increasingly popular in a broad range of machine learning tasks, ranging from anomaly detection and manifold learning to graph classification.
Persistent homology being the key approach in TDA provides a unique topological fingerprint of the data by assessing the evolution of various hidden patterns in the data as we vary a scale parameter. Current PH tools are limited to analyze the data by filtering with single parameter while in many applications, several relevant parameters are equally important to get a much finer information on the data. In this paper, we overcome this problem by introducing Effective Multidimensional Persistence (EMP) framework which enables to investigate the data by varying multiple scale parameters simultaneously. EMP framework provides a highly expressive summary of the data by integrating the multiple descriptor functions to the process successfully. EMP naturally adapts to many known single PH summaries and convert them into multidimensional summaries, for example, EMP Landscapes, EMP Silhouettes, EMP Images, and EMP Surfaces. These summaries deliver a multidimensional fingerprint of the data as matrices and arrays which are suitable for various machine learning models.
We apply EMP framework in graph classification tasks and observe that EMP boosts the performances of various single PH descriptors, and outperforms the most state-of-the-art methods on benchmark datasets. We further derive theoretical guarantees of the proposed EMP summary and prove the stability properties.

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