Probing Equivariance and Symmetry Breaking in Convolutional Networks

Sharvaree Vadgama (University of Amsterdam) · Mohammad Islam (University of Amsterdam, University of Amsterdam) · Domas Buracas (University of Pennsylvania) · Christian A Shewmake (Redwood Center, UC Berkeley) · Artem Moskalev (Johnson and Johnson) · Erik Bekkers (University of Amsterdam)
empirical studyequivariant modelsgenerationgeometric reference framesgroup-equivariancemodel selectionnon-equivariant modelsperformance trendspoint cloudsregressionrepresentation capacitysegmentationstructural priorssymmetry-breakingtask geometryunified group convolutional architecture

In this work, we explore the trade-offs of explicit structural priors, particularly group-equivariance. We address this through theoretical analysis and a comprehensive empirical study focusing on point clouds. To enable controlled and fair comparisons, we introduce \texttt{Rapidash}, a unified group convolutional architecture that allows for different variants of equivariant and non-equivariant models. Our results suggest that more constrained equivariant models outperform less constrained alternatives when aligned with the geometry of the task, and increasing representation capacity does not fully eliminate performance gaps. We see improved performance of models with equivariance and symmetry-breaking through tasks like segmentation, regression, and generation across diverse datasets. Explicit \textit{symmetry breaking} via geometric reference frames consistently improves performance, while \textit{breaking equivariance} through geometric input features can be helpful when aligned with task geometry. Our results provide task-specific performance trends that offer a more nuanced way for model selection. Code available at [github.com/Sharvaree/EquivarianceStudy](https://github.com/Sharvaree/EquivarianceStudy)