Aligning Network Equivariance with Data Symmetry: A Theoretical Framework and Adaptive Approach for Image Restoration

• URL: http://arxiv.org/abs/2605.13744v1
• Date: Wed, 13 May 2026 16:22:19 GMT
• Title: Aligning Network Equivariance with Data Symmetry: A Theoretical Framework and Adaptive Approach for Image Restoration
• Abstract summary: Image restoration is an inherently ill posed inverse problem.<n>We use non strict symmetry as a constraint to formulate the restoration inverse problem.<n>We show that the equivariance error of the optimal restoration operator is strictly bounded by the data symmetry error and the discretization mesh size.
• Score: 71.51826063983434
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Image restoration is an inherently ill posed inverse problem. Equivariant networks that embed geometric symmetry priors can mitigate this ill posedness and improve performance. However, current understanding of the relationship between network equivariance and data symmetry remains largely heuristic. Particularly for real world data with imperfect symmetry, existing research lacks a systematic theoretical framework to quantify symmetry, select transformation groups, or evaluate model data alignment. To bridge this gap, we conduct an analysis from an optimization perspective and formalize the intrinsic relationship among data symmetry priors, model equivariance, and generalization capability. Specifically, we propose for the first time a quantifiable definition of non strict symmetry at the dataset level (rather than sample level) and use it as a constraint to formulate the restoration inverse problem. We then show that the equivariance for restoration models can be naturally derived from this inverse problems incorporated the proposed symmetry constraints, and that the equivariance error of the optimal restoration operator is strictly bounded by the data symmetry error and the discretization mesh size. Furthermore, by analyzing the network's empirical risk, we demonstrate that aligning equivariance with data symmetry optimizes the bias variance trade off, minimizing the total expected risk. Guided by these insights, we propose a Sample Adaptive Equivariant Network that uses a hypernetwork and transformation learnable equivariant convolutions to dynamically align with each sample's inherent symmetry. Extensive experiments on super resolution, denoising, and deraining validate our theoretical findings and show significant superiority over standard baselines and traditional equivariant models. Our code and supplementary material are available at https://github.com/tanfy929/SA-Conv.

Related papers

• To Augment or Not to Augment? Diagnosing Distributional Symmetry Breaking [23.524227670982544]
  We propose a metric to quantify the amount of anisotropy, or symmetry-breaking, in a dataset.<n>We use it to uncover surprisingly high degrees of alignment in several benchmark point cloud datasets.
  arXiv  Detail & Related papers  (2025-10-01T18:26:33Z)
• Learning (Approximately) Equivariant Networks via Constrained Optimization [25.51476313302483]
  Equivariant neural networks are designed to respect symmetries through their architecture.<n>Real-world data often departs from perfect symmetry because of noise, structural variation, measurement bias, or other symmetry-breaking effects.<n>We introduce Adaptive Constrained Equivariance (ACE), a constrained optimization approach that starts with a flexible, non-equivariant model.
  arXiv  Detail & Related papers  (2025-05-19T18:08:09Z)
• Improving Equivariant Networks with Probabilistic Symmetry Breaking [9.164167226137664]
  Equivariant networks encode known symmetries into neural networks, often enhancing generalizations.<n>This poses an important problem, both (1) for prediction tasks on domains where self-symmetries are common, and (2) for generative models, which must break symmetries in order to reconstruct from highly symmetric latent spaces.<n>We present novel theoretical results that establish sufficient conditions for representing such distributions.
  arXiv  Detail & Related papers  (2025-03-27T21:04:49Z)
• Rao-Blackwell Gradient Estimators for Equivariant Denoising Diffusion [55.95767828747407]
  In domains such as molecular and protein generation, physical systems exhibit inherent symmetries that are critical to model.<n>We present a framework that reduces training variance and provides a provably lower-variance gradient estimator.<n>We also present a practical implementation of this estimator incorporating the loss and sampling procedure through a method we call Orbit Diffusion.
  arXiv  Detail & Related papers  (2025-02-14T03:26:57Z)
• Approximate Equivariance in Reinforcement Learning [35.04248486334824]
  We develop approximately equivariant algorithms in reinforcement learning.<n>Results show that the approximately equivariant network performs on par with exactly equivariant networks when exact symmetries are present.
  arXiv  Detail & Related papers  (2024-11-06T19:44:46Z)
• Symmetry Breaking and Equivariant Neural Networks [17.740760773905986]
  We introduce a novel notion of'relaxed equiinjection' We show how to incorporate this relaxation into equivariant multilayer perceptronrons (E-MLPs) The relevance of symmetry breaking is then discussed in various application domains.
  arXiv  Detail & Related papers  (2023-12-14T15:06:48Z)
• Learning Layer-wise Equivariances Automatically using Gradients [66.81218780702125]
  Convolutions encode equivariance symmetries into neural networks leading to better generalisation performance. symmetries provide fixed hard constraints on the functions a network can represent, need to be specified in advance, and can not be adapted. Our goal is to allow flexible symmetry constraints that can automatically be learned from data using gradients.
  arXiv  Detail & Related papers  (2023-10-09T20:22:43Z)
• Approximation-Generalization Trade-offs under (Approximate) Group Equivariance [6.368871731116769]
  The explicit incorporation of task-specific inductive biases through symmetry has emerged as a general design precept in the development of high-performance machine learning models.<n>We present general quantitative bounds that demonstrate how models capturing task-specific symmetries lead to improved generalization.<n>We examine the more general question of model mis-specification when the model symmetries don't align with the data symmetries.
  arXiv  Detail & Related papers  (2023-05-27T22:53:37Z)
• The Lie Derivative for Measuring Learned Equivariance [84.29366874540217]
  We study the equivariance properties of hundreds of pretrained models, spanning CNNs, transformers, and Mixer architectures. We find that many violations of equivariance can be linked to spatial aliasing in ubiquitous network layers, such as pointwise non-linearities. For example, transformers can be more equivariant than convolutional neural networks after training.
  arXiv  Detail & Related papers  (2022-10-06T15:20:55Z)
• Equivariance Discovery by Learned Parameter-Sharing [153.41877129746223]
  We study how to discover interpretable equivariances from data. Specifically, we formulate this discovery process as an optimization problem over a model's parameter-sharing schemes. Also, we theoretically analyze the method for Gaussian data and provide a bound on the mean squared gap between the studied discovery scheme and the oracle scheme.
  arXiv  Detail & Related papers  (2022-04-07T17:59:19Z)
• Learning Invariances in Neural Networks [51.20867785006147]
  We show how to parameterize a distribution over augmentations and optimize the training loss simultaneously with respect to the network parameters and augmentation parameters. We can recover the correct set and extent of invariances on image classification, regression, segmentation, and molecular property prediction from a large space of augmentations.
  arXiv  Detail & Related papers  (2020-10-22T17:18:48Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.