Learning with Holographic Reduced Representations

TL;DR

This paper revisits Holographic Reduced Representations (HRRs) to enable differentiable neural-symbolic learning, addressing numerical instability with a projection step, significantly improving concept retrieval and demonstrating effective multi-label classification.

Contribution

Introduces a projection step to stabilize HRRs in neural networks, enabling effective differentiable symbolic learning within deep architectures.

Findings

Concept retrieval improved over 100×

Effective multi-label classification achieved

Demonstrates viability of HRR-based neuro-symbolic learning

Abstract

Holographic Reduced Representations (HRR) are a method for performing symbolic AI on top of real-valued vectors by associating each vector with an abstract concept, and providing mathematical operations to manipulate vectors as if they were classic symbolic objects. This method has seen little use outside of older symbolic AI work and cognitive science. Our goal is to revisit this approach to understand if it is viable for enabling a hybrid neural-symbolic approach to learning as a differentiable component of a deep learning architecture. HRRs today are not effective in a differentiable solution due to numerical instability, a problem we solve by introducing a projection step that forces the vectors to exist in a well behaved point in space. In doing so we improve the concept retrieval efficacy of HRRs by over $100\times$. Using multi-label classification we demonstrate how to leverage the symbolic HRR properties to develop an output layer and loss function that is able to learn effectively, and allows us to investigate some of the pros and cons of an HRR neuro-symbolic learning approach. Our code can be found at https://github.com/NeuromorphicComputationResearchProgram/Learning-with-Holographic-Reduced-Representations