Product Manifold Representations for Learning on Biological Pathways

Machine learning models that embed graphs in non-Euclidean spaces have shown substantial benefits in a variety of contexts, but their application has not been studied extensively in the biological domain, particularly with respect to biological pathway graphs. Such graphs exhibit a variety of complex network structures, presenting challenges to existing embedding approaches. Learning high-quality embeddings for biological pathway graphs is important for researchers looking to understand the underpinnings of disease and train high-quality predictive models on these networks. In this work, we investigate the effects of embedding pathway graphs in non-Euclidean mixed-curvature spaces and compare against traditional Euclidean graph representation learning models. We then train a supervised model using the learned node embeddings to predict missing protein-protein interactions in pathway graphs. We find large reductions in distortion and boosts on in-distribution edge prediction performance as a result of using mixed-curvature embeddings and their corresponding graph neural network models. However, we find that mixed-curvature representations underperform existing baselines on out-of-distribution edge prediction performance suggesting that these representations may overfit to the training graph topology. We provide our mixed-curvature product GCN code at https://github.com/mcneela/Mixed-Curvature-GCN and our pathway analysis code at https://github.com/mcneela/Mixed-Curvature-Pathways.

Almost Equivariance via Lie Algebra Convolutions

Recently, the equivariance of models with respect to a group action has become an important topic of research in machine learning. However, imbuing an architecture with a specific group equivariance imposes a strong prior on the types of data transformations that the model expects to see. While strictly-equivariant models enforce symmetries, real-world data does not always conform to such strict equivariances, be it due to noise in the data or underlying physical laws that encode only approximate or partial symmetries. In such cases, the prior of strict equivariance can actually prove too strong and cause models to underperform on real-world data. Therefore, in this work we study a closely related topic, that of almost equivariance. We provide a definition of almost equivariance that differs from those extant in the current literature and give a practical method for encoding almost equivariance in models by appealing to the Lie algebra of a Lie group. Specifically, we define Lie algebra convolutions and demonstrate that they offer several benefits over Lie group convolutions, including being well-defined for non-compact groups. From there, we pivot to the realm of theory and demonstrate connections between the notions of equivariance and isometry and those of almost equivariance and almost isometry, respectively. We prove two existence theorems, one showing the existence of almost isometries within bounded distance of isometries of a general manifold, and another showing the converse for Hilbert spaces. We then extend these theorems to prove the existence of almost equivariant manifold embeddings within bounded distance of fully equivariant embedding functions, subject to certain constraints on the group action and the function class. Finally, we demonstrate the validity of our approach by benchmarking against datasets in fully equivariant and almost equivariant settings.