  Deep learning has achieved remarkable success in recent years. Central to its
success is its ability to learn representations that preserve task-relevant
structure. However, massive energy, compute, and data costs are required to
learn general representations. This paper explores Hyperdimensional Computing
(HDC), a computationally and data-efficient brain-inspired alternative. HDC
acts as a bridge between connectionist and symbolic approaches to artificial
intelligence (AI), allowing explicit specification of representational
structure as in symbolic approaches while retaining the flexibility of
connectionist approaches. However, HDC's simplicity poses challenges for
encoding complex compositional structures, especially in its binding operation.
To address this, we propose Generalized Holographic Reduced Representations
(GHRR), an extension of Fourier Holographic Reduced Representations (FHRR), a
specific HDC implementation. GHRR introduces a flexible, non-commutative
binding operation, enabling improved encoding of complex data structures while
preserving HDC's desirable properties of robustness and transparency. In this
work, we introduce the GHRR framework, prove its theoretical properties and its
adherence to HDC properties, explore its kernel and binding characteristics,
and perform empirical experiments showcasing its flexible non-commutativity,
enhanced decoding accuracy for compositional structures, and improved
memorization capacity compared to FHRR.
