A Probabilistic Approach to Learning the Degree of Equivariance in Steerable CNNs

2024-06-06

International Conference on Machine Learning

AI Summary

Steerable CNNs (SCNNs) suffer performance degradation when geometric symmetries are unknown or dynamically varying, due to overly restrictive equivariance constraints. Method: We propose a probabilistic equivariance modeling framework that quantifies, for each layer, the degree of equivariance to subgroups of compact groups via probability distributions over the group—parameterized efficiently using Fourier coefficients. The approach integrates group representation theory, Fourier analysis, and variational inference within the SCNN architecture, requiring no structural modifications and supporting both layer-wise independent and shared learning, alongside interpretability-promoting regularization. Contributions/Results: Our method achieves competitive accuracy on datasets with mixed symmetries. The learned probability distributions accurately reflect ground-truth equivariance patterns, demonstrating both empirical effectiveness and principled interpretability—enabling adaptive, data-driven equivariance without prior symmetry knowledge.

Technology Category

• Computer Vision: Representation Learning for Vision
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Steerable convolutional neural networks (SCNNs) enhance task performance by modelling geometric symmetries through equivariance constraints on weights. Yet, unknown or varying symmetries can lead to overconstrained weights and decreased performance. To address this, this paper introduces a probabilistic method to learn the degree of equivariance in SCNNs. We parameterise the degree of equivariance as a likelihood distribution over the transformation group using Fourier coefficients, offering the option to model layer-wise and shared equivariance. These likelihood distributions are regularised to ensure an interpretable degree of equivariance across the network. Advantages include the applicability to many types of equivariant networks through the flexible framework of SCNNs and the ability to learn equivariance with respect to any subgroup of any compact group without requiring additional layers. Our experiments reveal competitive performance on datasets with mixed symmetries, with learnt likelihood distributions that are representative of the underlying degree of equivariance.

Problem

• Learning optimal equivariance degree in steerable CNNs
• Handling unknown or varying symmetry constraints
• Modeling layer-wise and shared equivariance probabilistically

Innovation

• Probabilistic method learns equivariance degree in SCNNs
• Parameterizes equivariance as Fourier coefficient distribution
• Regularizes distributions for interpretable network equivariance