[Paper Review] Learning metrics for persistence-based summaries and applications for graph classification

Qi Zhao, Yusu Wang|arXiv (Cornell University)|Nov 1, 2019

Topological and Geometric Data Analysis32 citations

TL;DR

This paper proposes WKPI, a learnable weighted kernel for persistence diagrams that optimizes metric learning via labeled data to improve graph classification. By learning data-specific weight functions for persistence features, WKPI achieves state-of-the-art or competitive performance on benchmark graph datasets, outperforming prior methods in key cases.

ABSTRACT

Recently a new feature representation and data analysis methodology based on a topological tool called persistent homology (and its persistence diagram summary) has gained much momentum. A series of methods have been developed to map a persistence diagram to a vector representation so as to facilitate the downstream use of machine learning tools. In these approaches, the importance (weight) of different persistence features are usually pre-set. However often in practice, the choice of the weight-function should depend on the nature of the specific data at hand. It is thus highly desirable to learn a best weight-function (and thus metric for persistence diagrams) from labelled data. We study this problem and develop a new weighted kernel, called WKPI, for persistence summaries, as well as an optimization framework to learn the weight (and thus kernel). We apply the learned kernel to the challenging task of graph classification, and show that our WKPI-based classification framework obtains similar or (sometimes significantly) better results than the best results from a range of previous graph classification frameworks on a collection of benchmark datasets.

Motivation & Objective

• Address the limitation of fixed, pre-set weight functions in persistence diagram vectorization, which may not reflect data-specific feature importance.
• Develop a metric learning framework that adapts the weighting of persistence features based on labeled training data.
• Enable end-to-end optimization of the kernel function for persistence diagrams to enhance downstream machine learning performance.
• Apply the learned kernel to the challenging task of graph classification, improving accuracy and robustness.
• Demonstrate that learned metrics outperform hand-crafted or default weightings in real-world graph classification benchmarks.

Proposed method

• Propose WKPI, a weighted kernel that assigns learnable weights to persistence features based on their relevance to the learning task.
• Formulate an optimization framework to learn the optimal weight-function from labeled data, using a differentiable objective function.
• Integrate the learned kernel into a graph classification pipeline, leveraging persistence-based summaries as input representations.
• Use gradient-based optimization to update the weight parameters during training, enabling end-to-end learning of the metric.
• Construct persistence diagrams from graph data and map them to vector representations using the learned kernel for classification.
• Train the model using standard supervised learning protocols, with the kernel's weights updated via backpropagation to minimize classification error.

Experimental results

Research questions

• RQ1 Can a data-driven, learnable weight-function for persistence diagrams improve graph classification performance compared to fixed or heuristic weightings?
• RQ2 How does the learned metric in WKPI compare to existing kernel methods in terms of classification accuracy on benchmark graph datasets?
• RQ3 To what extent does metric learning from labeled data enhance the representational power of persistence diagrams in downstream tasks?
• RQ4 Can the proposed framework generalize across diverse graph classification benchmarks with varying topological complexity?
• RQ5 What is the impact of the optimization framework on the stability and convergence of the learned kernel?

Key findings

• The WKPI framework achieves similar or significantly better classification accuracy than state-of-the-art graph classification methods on multiple benchmark datasets.
• The learned weight-function consistently improves performance by adapting to the intrinsic topological structure of the data, outperforming fixed-weight baselines.
• The optimization framework successfully learns meaningful and discriminative weights for persistence features, enhancing the discriminative power of the kernel.
• The method demonstrates robustness across diverse graph types and topological complexities, indicating generalization capability.
• The results confirm that learning the metric from labeled data is critical for maximizing the utility of persistence diagrams in machine learning pipelines.