Blind Recovery of Signal Domains via Unsupervised Symmetry Discovery

A signal-processing perspective interprets data as discretized signals over geometric domains. When the underlying domain is inaccessible, we lose geometric priors, hindering interpretability and generalization. We propose an unsupervised framework for recovering signal domains by discovering symmetries of the data distribution. Our framework treats data samples as unknown linear measurements of continuous latent signals, and learns to recover them through a linear group-convolutional network. Assuming latent signals form a stationary Markov random field, we train the model by enforcing stationarity, minimizing total correlation, and maximizing information. We theoretically show that the proposed approach transforms an otherwise unidentifiable inverse problem into blind deconvolution, and we test the model on stochastic processes, Ising models, scrambled images, and biological neural recordings, revealing latent signals. Our results suggest a promising direction for unsupervised structure learning and blind inverse problems.