Bayesian 3D Steerable CNNs: Enabling Equivariance and Uncertainty Quantification Simultaneously

2026-06-13

AI Summary

This work proposes a Bayesian steerable convolutional neural network that unifies SE(3)-equivariant inductive bias with principled uncertainty quantification—a capability lacking in conventional steerable CNNs, which limits their deployment in high-stakes scenarios. By placing posterior distributions over the coefficients of steerable basis functions, the method constructs probabilistic 3D convolutional kernels and employs variational inference with Bayes-by-Backpropagation to optimize stochastic kernel parameters. This approach rigorously preserves SE(3) equivariance while disentangling epistemic and aleatoric uncertainties. Experiments demonstrate that under distribution shifts induced by additive Gaussian noise, the model achieves up to a 6.17% improvement in accuracy, an expected calibration error as low as 0.0263, and leverages uncertainty-guided decision-making to yield approximately 4% accuracy gains on 84% of test samples.

Abstract

Steerable convolutional neural networks (Steerable-CNNs) guarantee SE(3)-equivariance by parameterizing kernels as linear combinations of steerable basis functions, but their deterministic nature precludes uncertainty quantification - limiting their use in settings where confidence estimates are essential. We propose a Bayesian Steerable-CNN that places posterior distributions over the basis coefficients, yielding stochastic kernels while preserving equivariance exactly. The loss function of the model is obtained via variational inference and minimized by Bayes-by-Backpropagation. The framework admits a decomposition of predictive uncertainty into epistemic and aleatoric components. Empirically, the model attains competitive classification accuracy alongside an expected calibration error of 0.0263 and outperforms its deterministic counterpart by up to 6.17% under distributional shift induced by additive Gaussian noise. Furthermore, we leverage the model's uncertainty estimates to enhance its performance significantly, achieving a notable gain - approximately 4% higher accuracy across 84% of the test dataset. A statistically significant negative correlation between epistemic uncertainty and prediction error confirms that the learned posterior variance is semantically meaningful. The framework unifies Bayesian uncertainty quantification with the inductive bias of equivariant CNNs.

Problem

• Bayesian
• Steerable CNNs
• Equivariance
• Uncertainty Quantification
• SE(3)

Innovation

• Bayesian Steerable-CNN
• SE(3)-equivariance
• uncertainty quantification
• variational inference
• epistemic uncertainty