Traditional language models, adept at next-token prediction in text sequences, often struggle with transduction tasks between distinct symbolic systems, particularly when parallel data is scarce. Addressing this issue, we introduce \textit{symbolic autoencoding} ($\Sigma$AE), a self-supervised framework that harnesses the power of abundant unparallel data alongside limited parallel data. $\Sigma$AE connects two generative models via a discrete bottleneck layer and is optimized end-to-end by minimizing reconstruction loss (simultaneously with supervised loss for the parallel data), such that the sequence generated by the discrete bottleneck can be read out as the transduced input sequence. We also develop gradient-based methods allowing for efficient self-supervised sequence learning despite the discreteness of the bottleneck. Our results demonstrate that $\Sigma$AE significantly enhances performance on transduction tasks, even with minimal parallel data, offering a promising solution for weakly supervised learning scenarios.

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