Discrete Dictionary-based Decomposition Layer for Structured Representation Learning

TL;DR

A Discrete Dictionary-based Decomposition (D3) layer designed to enhance the decomposition capabilities of TPR-based models, which significantly improves the systematic generalization of various TPR-based models while requiring fewer additional parameters.

Abstract

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.

Figures (16)

• Figure 1: Overview of D3.D3 generates structured TPR representations by mapping input data to the nearest pre-learned symbolic features stored within discrete, learnable dictionaries. Each dictionary is linked explicitly to specific TPR components, such as roles, filler, and unbinding operators. Notably, D3 uses a shared dictionary configuration between the roles and unbinding operators. This figure illustrates, for example, that $\textit{role}_1$ and $\textit{unbind}_1$ share one dictionary, while $\textit{role}_2$ and $\textit{unbind}_2$ share another. $T$ denotes a superimposed representation that represents multiple objects.
• Figure 2: Test accuracy curve [%] on the SAR task for 10 seeds, with shadowed area indicating SD.
• Figure 3: The heatmap displays the cosine similarity between the generated representations during the discovery phase for the SAR task. We explore the similarity across different types of representations: (a) queries of roles, (b) codes of roles, and (c) the roles themselves.
• Figure 4: The heatmap displays the cosine similarity between the generated representations during the discovery phase (represented on the x-axis) and the inference phase (represented on the y-axis) for the SAR task. We explore the similarity across different types of representations: (a) queries of roles and unbinding operators, (b) codes of roles and unbinding operators, and (c) the roles and unbinding operators themselves.
• Figure 5: The heatmap visualizes the cosine similarity of the learned codebook features for the SAR task. There are two parts to each heatmap: (a) the similarity among codebook keys, denoted as $\{\textsf{k}_i\}_{i=1}^{N\text{code}}$, and (b) the similarity among codebook values, denoted as $\{\textsf{v}_i\}_{i=1}^{N\text{code}}$. For better visualization, the heatmap values are reordered to reflect the cluster of similar codebook keys.