  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
