Primary motivation in blind inverse problems is to recover signals of interest from corrupted observations without knowing the obfuscating mechanism. Blind deconvolution is a prominent approach when the corruption is convolutional, but it is not applicable when general linear transformations obfuscate the domain structure. In this work, we propose an unsupervised framework for recovering latent domains and signals by discovering symmetries of the data distribution. Our framework models observations as linear measurements of signals sampled from a latent random field, and optimizes a shallow group-convolutional network by imposing stationarity and locality regularization at the model output. The model learns a latent symmetry action and an appropriate filter, thereby mapping unstructured observations to a symmetry-based representation that reveals latent signals. Experiments on stochastic processes, Ising models, shuffled and bit-scrambled images, and neural recordings show that the method recovers latent domains and signals from unstructured observations, suggesting symmetry discovery as a new direction for unsupervised structure learning and blind inverse problems.

翻译：盲逆问题的主要目标是在未知混淆机制的情况下，从损坏的观测中恢复感兴趣的信号。当损坏表现为卷积形式时，盲反卷积是一种主流方法，但若一般线性变换混淆了域结构，该方法便不再适用。本文提出一种无监督框架，通过发现数据分布的对称性来恢复潜在域和信号。该框架将观测建模为从潜在随机场采样信号的线性测量值，并通过在模型输出端施加平稳性与局部性正则化，优化一个浅层群卷积网络。模型学习潜在对称性作用与合适的滤波器，从而将无结构的观测映射为基于对称性的表征，进而揭示潜在信号。在随机过程、伊辛模型、打乱与比特置乱图像以及神经记录上的实验表明，该方法能从无结构观测中恢复潜在域和信号，这揭示了对称性发现作为无监督结构学习与盲逆问题的新方向。