D-GRIL: End-to-End Topological Learning with 2-parameter Persistence (2406.07100v2)

Abstract: End-to-end topological learning using 1-parameter persistence is well-known. We show that the framework can be enhanced using 2-parameter persistence by adopting a recently introduced 2-parameter persistence based vectorization technique called GRIL. We establish a theoretical foundation of differentiating GRIL producing D-GRIL. We show that D-GRIL can be used to learn a bifiltration function on standard benchmark graph datasets. Further, we exhibit that this framework can be applied in the context of bio-activity prediction in drug discovery.

• The paper introduces D-Gril, leveraging a differentiable, piecewise affine Gril vectorization to reliably learn bifiltration functions.
• It employs stochastic sub-gradient descent under local Lipschitz continuity and Whitney stratifiability, ensuring almost sure convergence.
• Experimental results show that D-Gril enhances bio-activity prediction and graph classification by capturing richer topological features than classical methods.

End-to-End Topological Learning with 2-Parameter Persistence: A Theoretical and Experimental Exploration of D-Gril

Introduction

The paper introduces D-Gril, an enhancement of end-to-end topological learning using $2$-parameter persistence modules. This novel approach is built upon the $2$-parameter persistence vectorization technique called Gril, previously introduced. The paper establishes a comprehensive theoretical foundation for the differentiable extension of Gril, leading to the formulation of D-Gril, which can be utilized for learning bifiltration functions in various datasets. The framework is experimentally validated using both standard graph datasets and bio-activity prediction in drug discovery, demonstrating its utility over classical $1$-parameter persistence and other multiparameter persistence approaches.

Theoretical Contributions

Differentiability and Piecewise Affine Nature of Gril

The foundation of D-Gril relies on the properties of Gril which is shown to be piecewise affine. The authors demonstrate that Gril, a vectorization of $2$-parameter persistence, is piecewise affine concerning its input, the bifiltration function defined over the simplicial complexes. This is achieved by constructing an arrangement of hyperplanes in the parameter space, partitioning it into regions where the Gril vectorization is affine.

For a given set of sampled center points and bifiltration functions, the Gril vector is computed as a function of the generalized rank invariant over discrete $\ell$-worms. The arrangement of hyperplanes ensures that the relative ordering of simplex coordinates is fixed within each region, leading to the conclusion that Gril's behavior is consistent and predictable within these regions. The authors formalize this through a series of definitions and theorems, ultimately proving the piecewise affine nature of Gril and detailing the conditions for its differentiability.

Stochastic Sub-Gradient Descent

Building on the differentiability of Gril, the paper addresses the convergence of stochastic sub-gradient descent when applied to this transformation. Stochastic sub-gradient descent is shown to converge almost surely under the conditions of local Lipschitz continuity and Whitney stratifiability—properties guaranteed by the piecewise affine nature of Gril and the definability within an o-minimal structure.

The framework leverages results from previous works on stochastic sub-gradient descent, applying these to the setting of $2$-parameter persistence modules. This theoretical guarantee is crucial for the practical implementation of D-Gril, ensuring that the learning process for the bifiltration function is stable and convergent.

Practical Implications and Experimental Validation

Bio-Activity Prediction in Drug Discovery

One of the significant applications of D-Gril is in the context of bio-activity prediction in drug discovery. The paper elucidates that traditional molecular fingerprints often fail to capture the intricate topological and structural characteristics of molecules which influence their interaction with biological targets. By learning a bifiltration function through D-Gril, the framework captures richer topological information, leading to more informed predictions.

The authors demonstrate the efficacy of D-Gril on several bio-activity prediction datasets extracted from the ChEMBL database. The experimental results show that models augmented with D-Gril outperform traditional graph neural network (GNN) models and other multiparameter persistence methods in terms of classification accuracy. This underscores the practical utility of D-Gril in enhancing prediction performance by leveraging topological insights that are inherently captured by the bifiltration learning process.

General Graph Classification

Beyond bio-activity prediction, the paper extends the application of D-Gril to standard graph classification benchmarks such as MUTAG, PROTEINS, and IMDB-BINARY. The experiments consistently show that the learned bifiltration functions provide superior performance compared to predefined bifiltration functions and other multiparameter persistence methods.

The augmented models exhibit robust classification performance, indicating that the learned bifiltration functions adaptively capture relevant topological features that are conducive to better generalization. This adaptability and performance enhancement corroborate the theoretical advantages of utilizing $2$-parameter persistence within an end-to-end learning framework.

Future Directions

The exploration of end-to-end topological learning with $2$-parameter persistence opens several avenues for future research and development. Potential directions include:

1. Scalability and Efficiency: Investigating techniques to further optimize the computational efficiency of D-Gril, especially for large-scale datasets.
2. Extended Applications: Applying D-Gril to other domains such as neuroscience, material science, and sensor networks, where topological features may yield significant insights.
3. Refinement of Vectorization Techniques: Developing improved vectorization techniques for multiparameter persistence that can be seamlessly integrated into the D-Gril framework.
4. Theoretical Extensions: Extending the theoretical foundations to cover more complex topological structures and exploring the convergence properties under broader conditions.

Conclusion

The paper presents a rigorous theoretical approach to enhancing topological learning using $2$-parameter persistence, culminating in the D-Gril framework. Through comprehensive theoretical analysis and experimental validation, the authors establish the practical benefits of end-to-end learning of bifiltration functions for graph-based datasets. The results indicate substantial improvements in various application domains, particularly in bio-activity prediction for drug discovery, underscoring the potential of topological methods in advancing machine learning and data analysis.