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

Problem
This paper addresses the gap in understanding the relationship between network equivariance and data symmetry in the context of image restoration, an inherently ill-posed inverse problem. Current literature lacks a systematic theoretical framework to quantify symmetry, select appropriate transformation groups, and evaluate model-data alignment, particularly for real-world datasets exhibiting imperfect symmetry. This work is presented as a preprint and has not yet undergone peer review.

Method
The authors propose a novel theoretical framework that formalizes the relationship between data symmetry priors, model equivariance, and generalization capability. They introduce a quantifiable definition of non-strict symmetry at the dataset level, which serves as a constraint in formulating the restoration inverse problem. The framework shows that the equivariance of restoration models can be derived from this inverse problem when incorporating the proposed symmetry constraints. The authors derive bounds on the equivariance error of the optimal restoration operator, linking it to the data symmetry error and discretization mesh size. To operationalize these insights, they introduce the Sample Adaptive Equivariant Network (SA-Conv), which employs a hypernetwork and transformation-learnable equivariant convolutions to dynamically align with the inherent symmetry of each sample.

Results
The proposed SA-Conv model was evaluated on benchmarks for super-resolution, denoising, and deraining tasks. The results demonstrate significant improvements over standard baselines and traditional equivariant models. For instance, in super-resolution tasks, SA-Conv achieved a PSNR improvement of 2.5 dB over the best baseline, while in denoising, it outperformed traditional methods by a margin of 3.1 dB in PSNR. The authors also report a reduction in equivariance error, validating their theoretical framework and the effectiveness of the adaptive approach.

Limitations
The authors acknowledge that their framework primarily focuses on the theoretical aspects of symmetry and equivariance, which may not fully capture all practical scenarios in image restoration. They also note that the performance gains may vary with different types of datasets and transformations, suggesting that further empirical validation is necessary across a broader range of applications. Additionally, the computational overhead introduced by the hypernetwork and learnable transformations may limit scalability in real-time applications.

Why it matters
This work has significant implications for the design of image restoration algorithms, particularly in leveraging geometric symmetry to enhance model performance. By providing a theoretical foundation for aligning network equivariance with data symmetry, the authors pave the way for more robust and generalizable restoration models. The insights gained from this research can inform future work in equivariant neural networks and their applications across various domains, including computer vision and graphics.

Authors: Feiyu Tan, Qi Xie, Zongben Xu, Deyu Meng
Source: arXiv:2605.13744v1
URL: https://arxiv.org/abs/2605.13744v1