让神经网络学习具备平移缩放不变性的近端算子，提升去噪鲁棒性
Learning Affine-Equivariant Proximal Operators
- 设计可证明保持仿射等变性的神经网络近端算子
- 在分布外噪声和仿射偏移下仍保持优异去噪性能
- 适合需要结构化先验的信号重建与逆问题场景
近端算子在信号处理与机器学习中广泛应用，尤其用于求解不适定逆问题。近期工作提出可学习近端网络（LPNs），能够为数据驱动且可能非凸的正则项计算精确的近端算子。然而，在许多场景中，正则项及其对应近端算子需包含额外结构，如平移和缩放等变性。本文提出仿射等变可学习近端网络（AE-LPNs），通过神经网络参数化，能严格保证计算出的近端算子具有平移与缩放等变性。我们在合成构造案例与真实数据上验证了该方法，特别是在分布外噪声和仿射偏移条件下进行去噪任务。实验表明，该等变学习近端算子在远超训练分布的噪声分布与仿射变化下表现出更强鲁棒性，显著提升了学习近端算子的实际应用价值。
原文摘要 · Abstract (English)
Proximal operators are fundamental across many applications in signal processing and machine learning, including solving ill-posed inverse problems. Recent work has introduced Learned Proximal Networks (LPNs), providing parametric functions that compute exact proximals for data-driven and potentially non-convex regularizers. However, in many settings it is important to include additional structure to these regularizers--and their corresponding proximals--such as shift and scale equivariance. In this work, we show how to obtain learned functions parametrized by neural networks that provably compute exact proximal operators while being equivariant to shifts and scaling, which we dub Affine-Equivariant Learned Proximal Networks (AE-LPNs). We demonstrate our results on synthetic, constructive examples, and then on real data via denoising in out-of-distribution settings. Our equivariant learned proximals enhance robustness to noise distributions and affine shifts far beyond training distributions, improving the practical utility of learned proximal operators
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