Recent advances in unsupervised learning have highlighted the possibility of learning to reconstruct signals from noisy and incomplete linear measurements alone. These methods play a key role in medical and scientific imaging and sensing, where ground truth data is often scarce or difficult to obtain. However, in practice, measurements are not only noisy and incomplete but also quantized. Here we explore the extreme case of learning from binary observations and provide necessary and sufficient conditions on the number of measurements required for identifying a set of signals from incomplete binary data. Our results are complementary to existing bounds on signal recovery from binary measurements. Furthermore, we introduce a novel self-supervised learning approach, which we name SSBM, that only requires binary data for training. We demonstrate in a series of experiments with real datasets that SSBM performs on par with supervised learning and outperforms sparse reconstruction methods with a fixed wavelet basis by a large margin.

翻译：无监督学习的最新进展揭示了仅从含噪且不完整的线性测量中学习重建信号的可能性。这些方法在医学与科学成像及传感领域发挥着关键作用，因为这些领域中的真实数据往往稀缺或难以获取。然而在实际应用中，测量不仅存在噪声和不完整性，还涉及量化问题。本文探讨了从二元观测中学习的极端情况，并给出了从不完整二元数据中识别信号集所需测量次数的充要条件。我们的结果与现有二元测量信号恢复的界互为补充。此外，我们提出了一种新型自监督学习方法（命名为SSBM），该方法仅需二元数据即可完成训练。通过一系列真实数据集实验表明，SSBM的性能与监督学习相当，且大幅优于基于固定小波基的稀疏重建方法。