  Neuro-symbolic neural networks have been extensively studied to integrate
symbolic operations with neural networks, thereby improving systematic
generalization. Specifically, Tensor Product Representation (TPR) framework
enables neural networks to perform differentiable symbolic operations by
encoding the symbolic structure of data within vector spaces. However,
TPR-based neural networks often struggle to decompose unseen data into
structured TPR representations, undermining their symbolic operations. To
address this decomposition problem, we propose a Discrete Dictionary-based
Decomposition (D3) layer designed to enhance the decomposition capabilities of
TPR-based models. D3 employs discrete, learnable key-value dictionaries trained
to capture symbolic features essential for decomposition operations. It
leverages the prior knowledge acquired during training to generate structured
TPR representations by mapping input data to pre-learned symbolic features
within these dictionaries. D3 is a straightforward drop-in layer that can be
seamlessly integrated into any TPR-based model without modifications. Our
experimental results demonstrate that D3 significantly improves the systematic
generalization of various TPR-based models while requiring fewer additional
parameters. Notably, D3 outperforms baseline models on the synthetic task that
demands the systematic decomposition of unseen combinatorial data.
