GRIL: A $2$-parameter Persistence Based Vectorization for Machine Learning

TL;DR

This paper introduces GRIL, a novel vectorization method for 2-parameter persistence modules in Topological Data Analysis, enhancing machine learning models like GNNs by capturing richer topological features.

Contribution

The paper proposes a new vector representation called GRIL for 2-parameter persistence modules, which is stable, differentiable, and easily integrated into machine learning models.

Findings

GRIL improves GNN performance on graph datasets.

Efficient algorithm for computing GRIL vectors.

Comparison shows GRIL captures more topological features.

Abstract

$1$-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation power of deep learning models, such as Graph Neural Networks (GNNs). To enrich the representations of topological features, here we propose to study $2$-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vector representation called Generalized Rank Invariant Landscape (GRIL) for $2$-parameter persistence modules. We show that this vector representation is $1$-Lipschitz stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vector representation efficiently. We also test our methods on synthetic and benchmark graph datasets, and compare the results with previous vector representations of $1$-parameter and $2$-parameter persistence modules. Further, we augment GNNs with GRIL features and observe an increase in performance indicating that GRIL can capture additional features enriching GNNs. We make the complete code for the proposed method available at https://github.com/soham0209/mpml-graph.