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

arXiv: https://arxiv.org/abs/2606.15479 · HTML: https://mlsanity.com/paper/2606.15479/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 28 references that are in the map.
- First posted on arXiv 13 Jun 2026.

## Builds on (most cited first, 21 of 21)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [R'enyi Divergence Variational Inference](https://mlsanity.com/paper/1602.02311/) (2016)
- [Sampling-free Epistemic Uncertainty Estimation Using Approximated Variance Propagation](https://mlsanity.com/paper/1908.00598/) (2019)
- [A Comprehensive guide to Bayesian Convolutional Neural Network with Variational Inference](https://mlsanity.com/paper/1901.02731/) (2019)
- [f-Divergence Variational Inference](https://mlsanity.com/paper/2009.13093/) (2020)
- [Constraining Variational Inference with Geometric Jensen-Shannon Divergence](https://mlsanity.com/paper/2006.10599/) (2020)
- [Explanation and Use of Uncertainty Quantified by Bayesian Neural Network Classifiers for Breast Histopathology Images](https://mlsanity.com/paper/2010.12575/) (2020)
- [A Probabilistic Approach to Learning the Degree of Equivariance in Steerable CNNs](https://mlsanity.com/paper/2406.03946/) (2024)
- [Jensen-Shannon Divergence Based Novel Loss Functions for Bayesian Neural Networks](https://mlsanity.com/paper/2209.11366/) (2022)

## Abstract (arXiv)

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.