Gaussian Processes on Cellular Complexes (2311.01198v2)

Abstract: In recent years, there has been considerable interest in developing machine learning models on graphs to account for topological inductive biases. In particular, recent attention has been given to Gaussian processes on such structures since they can additionally account for uncertainty. However, graphs are limited to modelling relations between two vertices. In this paper, we go beyond this dyadic setting and consider polyadic relations that include interactions between vertices, edges and one of their generalisations, known as cells. Specifically, we propose Gaussian processes on cellular complexes, a generalisation of graphs that captures interactions between these higher-order cells. One of our key contributions is the derivation of two novel kernels, one that generalises the graph Mat\'ern kernel and one that additionally mixes information of different cell types.

• The paper introduces Gaussian processes on cellular complexes to capture higher-order interactions beyond pairwise relations.
• It presents two novel kernels—the cellular Matérn kernel and a reaction-diffusion kernel—for modeling signals across arbitrary cell types.
• The framework enhances uncertainty quantification in predictions, offering advancements for applications in complex systems such as social networks and computational biology.

Gaussian Processes on Cellular Complexes

The paper "Gaussian Processes on Cellular Complexes" presents a novel framework for leveraging Gaussian processes (GPs) on structures that extend beyond traditional graphs. In scenarios where interactions among entities are not merely dyadic but involve multiple entities, an enriched model becomes necessary. The authors address these needs by proposing GPs on cellular complexes, which are topological spaces capable of accommodating higher-order interactions.

Key Contributions

The primary contribution is the development of Gaussian processes on cellular complexes, which allow modeling of interactions encapsulated in higher-order cells such as vertices, edges, and volumes. This is achieved by:

1. Introducing Novel Kernels: Two new kernels are proposed:
   • A cellular Matérn kernel that generalizes the graph Matérn kernel for modeling signals across arbitrary cell types.
   • A reaction-diffusion kernel that utilizes the Dirac operator to integrate information across various cell types.
2. Addressing Limitations of Graphs: While graphs can efficiently model relations between pairs of vertices, they fall short in representing polyadic interactions. Cellular complexes overcome these limitations, opening up applications in areas like social network dynamics and neuron interactions.
3. Bringing Uncertainty Quantification: Unlike graph neural networks, the proposed GPs naturally incorporate uncertainty, an essential aspect for decision-making applications.

Implications of Research

The proposed framework has both theoretical and practical implications. Theoretically, it extends the modeling capabilities of GPs beyond graphs to capture complex interactions in data. Practically, it enhances applications in fields such as computational biology and climate modeling, where understanding higher-order interactions is crucial.

Numerical Results and Key Findings

The experiments conducted in this research demonstrate the proficiency of the proposed models in two main scenarios:

• Directed Edge Prediction: Applying the edge-based Matérn GP to ocean current data shows that CC-GPs not only manage directionality of flow but also offer improved uncertainty quantification over traditional graph GPs.
• Signal Mixing: The comparison between the cellular Matérn and reaction-diffusion kernels illustrates the superiority of the latter in scenarios involving strong correlations between different cell types. The RD kernel models the interactions effectively, leading to enhanced prediction accuracy.

The numerical results underscore the benefits of using cellular structures and specialized kernels, marking a significant step forward in modeling complex systems.

Future Directions

The framework presented paves the way for several potential future developments:

• Scalability and Efficiency: Computational costs, particularly concerning eigendecomposition, could be prohibitive for large networks. Future work could explore scalable approximations and optimizations.
• Augmenting with Neural Networks: Hybrid models incorporating the flexibility of neural networks with the uncertainty quantification of GPs are promising avenues for further exploration.
• Broadening Application Domains: Beyond the demonstrated domains, the framework is well-suited to any field involving high-dimensional, structured data.

Conclusion

This paper successfully ventures into extending Gaussian processes to model complex interactions using cellular complexes. By addressing the limitations inherent to graph structures in modeling higher-order interactions, and introducing powerful kernels, it opens new paths for theoretical inquiry and practical application across diverse fields. The translation of these theoretical constructs into effective computational tools marks a notable contribution to the field of machine learning and artificial intelligence.